R. Van Peski Van Peski, R.

<EMAIL_ADDRESS>
# Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

Roger Van Peski  1  11affiliationmark:  Department of Mathematics,
Massachusetts Institute of Technology, USA

(1 Month 20XX; 11 Month 20XX; 21 Month 20XX; A. Editor)

###### Abstract

We prove that the boundary of the Hall-Littlewood $t$-deformation of the
Gelfand-Tsetlin graph is parametrized by infinite integer signatures,
extending results of Gorin [23] and Cuenca [15] on boundaries of related
deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$
we use this to recover results of Bufetov-Qiu [12] and Assiotis [1] on
infinite $p$-adic random matrices, placing them in the general context of
branching graphs derived from symmetric functions.

Our methods rely on explicit formulas for certain skew Hall-Littlewood
polynomials. As a separate corollary to these, we obtain a simple expression
for the joint distribution of the cokernels of products
$A_{1},A_{2}A_{1},A_{3}A_{2}A_{1},\ldots$ of independent Haar-distributed
matrices $A_{i}$ over $\mathbb{Z}_{p}$, generalizing the explicit formula for
the classical Cohen-Lenstra measure.

###### Contents

  1. 1 Introduction
  2. 2 Hall-Littlewood polynomials
  3. 3 Principally specialized skew Hall-Littlewood polynomials
  4. 4 The $t$-deformed Gelfand-Tsetlin graph and its boundary
  5. 5 Infinite $p$-adic random matrices and corners
  6. 6 Ergodic decomposition of $p$-adic Hua measures
  7. 7 Products of finite $p$-adic random matrices
  8. A Markov dynamics on the boundary

##  1 Introduction

###  1.1 Hall-Littlewood polynomials.

The classical _Hall-Littlewood polynomials_
$P_{\lambda}(x_{1},\ldots,x_{n};t)$ are a family of symmetric polynomials in
variables $x_{1},\ldots,x_{n}$, with an additional parameter $t$, indexed by
weakly decreasing sequences of nonnegative integers
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n})$ (called
nonnegative signatures). They reduce to Schur polynomials at $t=0$ and
monomial symmetric polynomials at $t=1$, and play key roles in geometry,
representation theory, and algebraic combinatorics. For this work, the most
relevant role is that for $t=1/p$, $p$ prime, they are intimately related to
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$-spherical functions on
$\operatorname{GL}_{n}(\mathbb{Q}_{p})$ [29, Chapter V], and consequently are
important in $p$-adic random matrix theory [39].

Explicitly they are defined by

$P_{\lambda}(x_{1},\ldots,x_{n};t):=\frac{1}{v_{\lambda}(t)}\sum_{\sigma\in
S_{n}}\sigma\left(x_{1}^{\lambda_{1}}\cdots x_{n}^{\lambda_{n}}\prod_{1\leq
i<j\leq n}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right),$ (1.1)

where $\sigma$ acts by permuting the variables and $v_{\lambda}(t)$ is the
normalizing constant such that the $x_{1}^{\lambda_{1}}\cdots
x_{n}^{\lambda_{n}}$ term has coefficient $1$. As with other families of
symmetric functions one may define _skew Hall-Littlewood polynomials_
$P_{\lambda/\mu}$ in terms of two nonnegative signatures $\lambda,\mu$ of
lengths $n,k$ by

$P_{\lambda}(x_{1},\ldots,x_{n};t)=\sum_{\mu\in\operatorname{\mathbb{GT}}_{k}}P_{\lambda/\mu}(x_{1},\ldots,x_{n-k};t)P_{\mu}(x_{n-k+1},\ldots,x_{n};t),$
(1.2)

where $\operatorname{\mathbb{GT}}_{k}$ is the set of signatures of length $k$.

###  1.2 Branching graphs from Hall-Littlewood polynomials.

In 1976, Voiculescu [42] classified the characters of the infinite unitary
group $U(\infty)$, defined as the inductive limit of the chain $U(1)\subset
U(2)\subset\ldots$. This was later shown to be equivalent to earlier results
by Aissen, Edrei, Schoenberg and Whitney, stated without reference to
representation theory. A similar story unfolded for the infinite symmetric
group $S_{\infty}$ [25, 40], related to the classical Thoma theorem [37]. See
[8, §1.1] and the references therein for a more detailed exposition of both.

In later works such as [41, 32] the result for $U(\infty)$ was recast in terms
of classifying the boundary of the so-called _Gelfand-Tsetlin branching graph_
, defined combinatorially in terms of Schur polynomials. This led to natural
generalizations to other branching graphs defined in terms of degenerations of
Macdonald polynomials $P_{\lambda}(x_{1},\ldots,x_{n};q,t)$, which feature two
parameters $q,t$ and specialize to Hall-Littlewood polynomials when $q=0$; see
[8, 15, 23, 32, 35]. In special cases these combinatorial results take on
additional significance in representation theory and harmonic analysis; the
Schur case was already mentioned, and two other special cases of the result of
[32] for the Jack polynomial case specialize to statements about the infinite
symmetric spaces $U(\infty)/O(\infty)$ and $U(2\infty)/Sp(\infty)$. For the
Young graph, the boundary of its natural Hall-Littlewood deformation was
conjectured in equivalent form in [26], proved in [30], and used to deduce
results on infinite matrices over finite fields in [16]. Surprisingly,
however, the boundary of the Hall-Littlewood deformation of the Gelfand-
Tsetlin graph has not previously been carried out, despite the fact that the
appearance of Hall-Littlewood polynomials in harmonic analysis on $p$-adic
groups suggests interpretations beyond the purely combinatorial setting.

Let us describe the setup of the Hall-Littlewood branching graph; we refer to
[9, Chapter 7] for an expository account of the general formalism of graded
graphs and their boundaries. Let
$\operatorname{\mathbb{GT}}_{n}=\\{(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}:\lambda_{1}\geq\ldots\geq\lambda_{n}\\}$
be the set of _integer signatures of length $n$_, not necessarily nonnegative.
Allowing $\lambda$ to be an arbitrary signature, (1.1) yields a symmetric
‘Hall-Littlewood Laurent polynomial’ which we also denote $P_{\lambda}$. Let
$\mathscr{G}_{t}$ be the weighted graph with vertices

$\bigsqcup_{n\geq 1}\operatorname{\mathbb{GT}}_{n}$

and edges between
$\lambda\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{n+1}$
with weights

$L_{n}^{n+1}(\mu,\lambda):=P_{\mu/\lambda}(t^{n};t)\frac{P_{\lambda}(1,\ldots,t^{n-1};t)}{P_{\mu}(1,\ldots,t^{n};t)},$
(1.3)

known as _cotransition probabilities_. These cotransition probabilities are
stochastic by (1.2), so any probability measure on
$\operatorname{\mathbb{GT}}_{n+1}$ induces another probability measure on
$\operatorname{\mathbb{GT}}_{n}$. A sequence of probability measures
$(M_{n})_{n\geq 1}$ which is consistent under these maps is called a _coherent
system_. As these form a simplex, understanding coherent systems reduces to
understanding the extreme points, called the _boundary_ of the branching
graph. Our first main result is an explicit description of the boundary of
$\mathscr{G}_{t}$. Here $\nu_{x}^{\prime}=\\#\\{i:\nu_{i}\geq x\\}$,
$(a;t)_{n}=\prod_{i=1}^{n}(1-at^{i-1})$ is the $t$-Pochhammer symbol,

$\begin{bmatrix}a\\\
b\end{bmatrix}_{t}=\frac{(t;t)_{a}}{(t;t)_{b}(t;t)_{a-b}}$

is the $t$-binomial coefficient, and we let
$\operatorname{\mathbb{GT}}_{\infty}$ be the set of weakly decreasing tuples
of integers $(\mu_{1},\mu_{2},\ldots)$.

thmboundary For any $t\in(0,1)$, the boundary of $\mathscr{G}_{t}$ is
naturally in bijection with $\operatorname{\mathbb{GT}}_{\infty}$. Under this
bijection $\mu\in\operatorname{\mathbb{GT}}_{\infty}$ corresponds to the
coherent system $(M^{\mu}_{n})_{n\geq 1}$ defined explicitly by

$M_{n}^{\mu}(\lambda):=(t;t)_{n}\prod_{x\in\mathbb{Z}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}\begin{bmatrix}\mu_{x}^{\prime}-\lambda_{x+1}^{\prime}\\\
\lambda_{x}^{\prime}-\lambda_{x+1}^{\prime}\end{bmatrix}_{t}$

for $\lambda\in\operatorname{\mathbb{GT}}_{n}$.

We note that the product over $x\in\mathbb{Z}$ in fact has only finitely many
nontrivial terms. The fact that the extreme measures have simple explicit
formulas is unusual for results of this type–usually, the measures are
characterized implicitly by certain generating functions.

The proof in Section 4 follows the general outline of the so-called _Vershik-
Kerov ergodic method_ , as do those of many related results mentioned above.
One of the closest works to our setting is [23], which studies the Schur
analogue with edge weights

$s_{\mu/\lambda}(t^{n})\frac{s_{\lambda}(1,\ldots,t^{n-1})}{s_{\mu}(1,\ldots,t^{n})}$

for $t\in(0,1)$, where $s_{\lambda}(x)$ is the Schur polynomial. The boundary
is shown to be naturally in bijection with
$\operatorname{\mathbb{GT}}_{\infty}$ as in our case111Our $t$ corresponds to
the $q^{-1}$ in the notation [23]. The setting of [23] actually corresponds to
$t>1$, and the boundary corresponds to infinite _increasing_ tuples of
integers, but this statement is equivalent to ours upon interchanging
signatures with their negatives–see the comment after Theorem 1.1 in [23]..

The boundary classification results of [23] are generalized in [15] to the
Macdonald case with cotransition probabilities

$P_{\mu/\lambda}(t^{n};q,t=q^{k})\frac{P_{\lambda}(1,\ldots,t^{n-1};q,t=q^{k})}{P_{\mu}(1,\ldots,t^{n};q,t=q^{k})}$
(1.4)

for any $k\in\mathbb{N}$, and the boundary is again identified with
$\operatorname{\mathbb{GT}}_{\infty}$; when $k=1$ this reduces to the result
of [23]. We do not see how Section 1.2 could be accessed by the methods of
[15] or the newer work [35], which treats the related _Extended Gelfand-
Tsetlin graph_ with weights coming from Macdonald polynomials with arbitrary
$q,t\in(0,1)$. Instead, we rely on explicit expressions, 3.2 and 3.3, for the
skew Hall-Littlewood polynomials appearing in (1.3). This means that Section
1.2 gives explicit formulas for the extreme coherent measures, while in
previous works they were defined implicitly by certain generating functions.

###  1.3 Ergodic measures on infinite $p$-adic random matrices.

In the special case $t=1/p$, the purely combinatorial results on Hall-
Littlewood polynomials have consequences in $p$-adic random matrix theory, and
we may deduce results of [12, 1] from Section 1.2 above. We refer to Section 7
for basic background on the $p$-adic integers $\mathbb{Z}_{p}$ and $p$-adic
field $\mathbb{Q}_{p}$. The group
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$
acts on $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ by left- and right
multiplication, and the orbits of this action on nonsingular matrices are
parametrized by the set $\operatorname{\overline{\mathbb{GT}}}_{\min(m,n)}$ of
‘extended’ signatures with parts allowed to be equal to $-\infty$. Explicitly,
for any $A\in\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p}),n\leq m$ there
exist
$U\in\operatorname{GL}_{n}(\mathbb{Z}_{p}),V\in\operatorname{GL}_{m}(\mathbb{Z}_{p})$
such that

$UAV=\operatorname{diag}_{n\times
m}(p^{-\lambda_{1}},\ldots,p^{-\lambda_{n}})$

for some $\lambda\in\operatorname{\overline{\mathbb{GT}}}_{n}$, where we take
$p^{\infty}=0$ by convention. The extended signature $\lambda$ is unique, and
we refer to the $\lambda_{i}$ as the _singular numbers_ of $A$ and write
$\operatorname{SN}(A)=\lambda\in\operatorname{\overline{\mathbb{GT}}}_{n}$.

For fixed $n\leq m$, the
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$
bi-invariant measures on $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ are
all convex combinations of those parametrized by
$\operatorname{\overline{\mathbb{GT}}}_{n}$ via $U\operatorname{diag}_{n\times
m}(p^{\lambda_{1}},\ldots,p^{\lambda_{n}})V$ with $U,V$ distributed by the
Haar measures on
$\operatorname{GL}_{n}(\mathbb{Z}_{p}),\operatorname{GL}_{m}(\mathbb{Z}_{p})$
respectively. One may define $\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$ as a
direct limit of the system

$\operatorname{GL}_{1}(\mathbb{Z}_{p})\subset\operatorname{GL}_{2}(\mathbb{Z}_{p})\subset\ldots$

and it is natural to ask for the extension of this result to infinite
matrices, i.e. for the extreme points in the set of
$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$ bi-invariant measures on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$. This problem was
previously solved in [12], which gave an explicit family of measures in
bijection with $\operatorname{\overline{\mathbb{GT}}}_{\infty}$. We give a new
proof that the extreme measures are naturally parametrized by
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ in Section 1.3 below.

thmrecoverBQ The set of extreme
$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})\times\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$-invariant
measures on $\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ is
naturally in bijection with $\operatorname{\overline{\mathbb{GT}}}_{\infty}$.
Under this bijection, the measure $E_{\mu}$ corresponding to
$\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ is the unique measure
such that its $n\times m$ truncations are distributed by the unique
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$-invariant
measure on $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ with singular
numbers distributed according to the measure $M^{\mu}_{m,n}$ defined in 5.4 in
the case $t=1/p$.

Our proof goes by deducing this parametrization by
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ from an augmented version of
the parametrization by $\operatorname{\mathbb{GT}}_{\infty}$ appearing in
Section 1.2. The key fact which relates the random matrix setting to the
purely combinatorial setting is a result, stated later as 5.2, which was
proven originally in [39]. This result relates the distribution of singular
numbers of a $p$-adic matrix after removing a row or column to the
cotransition probabilities (1.3).

We note that while Hall-Littlewood polynomials are not mentioned by name in
[12], it should be possible to extrapolate many of their Fourier analytic
methods to statements about Hall-Littlewood polynomials at general $t$. Our
methods, which are based on explicit formulas for certain skew Hall-Littlewood
polynomials, nonetheless differ substantially from those of [12] in a manner
which is not merely linguistic. Let us also be clear that while both Section
1.3 and [12] show that the extreme measures are parametrized by
$\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$, it is not obvious from
the descriptions that the measures corresponding to a given
$\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ under [12] and Section
1.3 are in fact the same. A separate argument, assuming the result of [12], is
required to prove that the two parametrizations by
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ match, see 5.5. This
additionally provides a computation of the distribution of singular numbers of
finite corners of matrices drawn from the measures in [12], which is new. We
refer to 9 for more detail on the differences between Section 1.3 and [12,
Theorem 1.3], in particular an explanation of how our results carry over to a
general non-Archimedean local field as is done in [12]. We mention also that
the other main result of [12] is a classification of the extreme measures on
infinite symmetric matrices
$\\{A\in\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p}):A^{T}=A\\}$
invariant under $\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$; it would be
interesting to have an analogous Hall-Littlewood proof of this result as well,
see 10 for further discussion of possible strategy and difficulties.

######  Remark 1.

In addition to [32], another work somewhat similar in spirit to Section 1.2
and Section 1.3 is [2]. This work finds the boundary of a certain branching
graph defined via multivariate Bessel functions–another degeneration of
Macdonald polynomials–and related to $\beta$-ensembles at general $\beta$. In
the classical values $\beta=1,2,4$ this recovers results on branching graphs
coming from random matrix theory. Results such as ours in terms of Hall-
Littlewood polynomials may be regarded as extrapolations of $p$-adic random
matrix theory to arbitrary real $p>1$ in the same way $\beta$-ensembles
extrapolate classical random matrix theory to real $\beta>0$, see also [39,
Remark 13].

###  1.4 Ergodic decompositions of $p$-adic Hua measures.

For finite random matrices over $\mathbb{Q}_{p}$ or $\mathbb{C}$, one wishes
to compute the distribution of singular numbers, singular values or
eigenvalues of certain distinguished ensembles such as the classical GUE,
Wishart and Jacobi ensembles (over $\mathbb{C}$), or the additive Haar measure
over $\mathbb{Z}_{p}$. The infinite-dimensional analogue of this problem is to
compute how distinguished measures on infinite matrices decompose into extreme
points, which correspond to ergodic measures. Such a decomposition is given by
a probability measure on the space of ergodic measures, which in our case
corresponds to a probability measure on
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$.

One such distinguished family of measures on $p$-adic matrices is given by the
_$p$ -adic Hua measures_ defined in [31], which are analogues of the complex
Hua-Pickrell measures222See [7], which coined the term for these measures, for
an historical discussion of these measures and summary of the contents of the
earlier works [24, 36].. There is a $p$-adic Hua measure
$\mathbb{M}_{n}^{(s)}$ on $\operatorname{Mat}_{n\times n}(\mathbb{Q}_{p})$ for
each $n\in\mathbb{Z}_{\geq 1},s\in\mathbb{R}_{>-1}$, which is defined by an
explicit density with respect to the underlying additive Haar measure on
$\operatorname{Mat}_{n\times n}(\mathbb{Q}_{p})$, see 21. A motivating
property of these measures is that they are consistent under taking corners,
and hence define a measure $\mathbb{M}_{\infty}^{(s)}$ on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$. The decomposition
of this measure into ergodic measures on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ was computed
recently in [1], and we reprove the result using the aforementioned relation
between $p$-adic matrix corners and the Hall-Littlewood branching graph
$\mathscr{G}_{t}$. Below $E_{\mu}$ is as in Section 1.3, $\mathbb{Y}$ is the
set of integer partitions, $Q_{\lambda}$ is the dual normalization of the
Hall-Littlewood symmetric function, and the normalizing constant
$\Pi(1,\ldots;u,\ldots)$ is the so-called _Cauchy kernel_ –see Section 2 for
precise definitions.

thmrecoverA Fix a prime $p$ and real parameter $s>-1$, and let $t=1/p$ and
$u=p^{-1-s}$. Then the infinite $p$-adic Hua measure
$\mathbb{M}_{\infty}^{(s)}$ decomposes into ergodic measures according to

$\mathbb{M}_{\infty}^{(s)}=\sum_{\mu\in\mathbb{Y}}\frac{P_{\mu}(1,t,\ldots;t)Q_{\mu}(u,ut,\ldots;t)}{\Pi(1,\ldots;u,\ldots)}E_{\mu}$
(1.5)

where $E_{\mu}$ is as defined in Section 1.3.

The key ingredient in the original proof of Section 1.4 given previously in
[1] is a certain Markov chain which generates the finite Hua measures
$\mathbb{M}_{n}^{(s)}$, and which was guessed from Markov chains appearing in
similar settings [19]. The arguments there did not use Hall-Littlewood
polynomials, but the limiting measure on
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ which describes the ergodic
decomposition was observed in [1] to be the so-called Hall-Littlewood measure
in (1.5), by matching explicit formulas. From our perspective, by contrast,
the fact that this measure is a Hall-Littlewood measure is natural and is key
to the proof.

###  1.5 From Hall-Littlewood polynomials to cokernels of products of $p$-adic
random matrices.

In another direction, random $p$-adic matrices have been subject to much
activity in arithmetic statistics going back to the 1983 conjectures of Cohen
and Lenstra [14] on class groups of quadratic imaginary number fields, and
their interpretation via random matrices in [18]. These works interpret the
singular numbers of a random matrix
$A\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})$ as specifying a random abelian
$p$-group: if $\operatorname{SN}(A)=-\lambda$ with $\lambda_{n}\geq 0$, then
viewing $A$ as a map $\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}^{n}$ one has

$\operatorname{coker}(A)=\mathbb{Z}_{p}^{n}/\text{Im}(A)\cong\bigoplus_{i=1}^{n}\mathbb{Z}/p^{\lambda_{i}}\mathbb{Z}=:G_{\lambda}(p).$

For $A_{n}\in\operatorname{Mat}_{n\times n}(\mathbb{Z}_{p})$ with iid entries
distributed according to the additive Haar measure on $\mathbb{Z}_{p}$, the
result of [18] implies

$\lim_{n\to\infty}\Pr(\operatorname{coker}(A_{n})\cong
G_{\lambda}(p))=\frac{1}{Z}P_{\lambda}(1,t,\ldots;t)Q_{\lambda}(t,t^{2},\ldots;t)=\frac{1}{Z}|\operatorname{Aut}(G_{\lambda}(p))|^{-1}$
(1.6)

where $t=1/p$ and $Z=\Pi(1,t,\ldots;t,t^{2},\ldots)$ is a normalizing
constant. For odd $p$ this distribution was conjectured to describe the
$p$-torsion parts of class groups random quadratic imaginary number fields
ordered by discriminant, and is often called the Cohen-Lenstra distribution
[14].

The next result generalizes the finite-$n$ version of (1.6) to arbitrary
products of independent additive Haar matrices. Here
$n(\lambda):=\sum_{i=1}^{n}(i-1)\lambda_{i}$ for
$\lambda\in\operatorname{\mathbb{GT}}_{n}$.

######  Theorem 1.1.

Let $t=1/p$, fix $n\geq 1$, and let $A_{i}$ be iid $n\times n$ matrices with
iid entries distributed by the additive Haar measure on $\mathbb{Z}_{p}$. Then
the joint distribution of
$\operatorname{coker}(A_{1}),\operatorname{coker}(A_{2}A_{1}),\ldots$ is given
by

$\displaystyle\begin{split}&\Pr(\operatorname{coker}(A_{i}\cdots A_{1})\cong
G_{\lambda(i-1)}(p)\text{ for all }i=1,\ldots,k)\\\
&=(t;t)_{n}^{k}t^{n(\lambda(k))}\prod_{1\leq i\leq
k}\prod_{x\in\mathbb{Z}}t^{\binom{\lambda(i)_{x}^{\prime}-\lambda(i-1)_{x}^{\prime}+1}{2}}\begin{bmatrix}\lambda(i)_{x}^{\prime}-\lambda(i-1)_{x+1}^{\prime}\\\
\lambda(i)_{x}^{\prime}-\lambda(i)_{x+1}^{\prime}\end{bmatrix}_{t}\end{split}$
(1.7)

for any $k$ and
$\lambda(1),\ldots,\lambda(k)\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$,
where we take $\lambda(0)=(0,\ldots,0)$ in (1.7).

Note that the product over $x\in\mathbb{Z}$, which may appear uninviting, in
fact has only finitely many nontrivial terms. As a special case one obtains
the prelimit version of (1.6), due to [18]: for
$A\in\operatorname{Mat}_{n}(\mathbb{Z}_{p})$ with iid additive Haar entries,

$\Pr(\operatorname{coker}(A)\cong
G_{\lambda}(p))=t^{2n(\lambda)+|\lambda|}\frac{(t;t)_{n}^{2}}{\prod_{i\geq
0}(t;t)_{m_{i}(\lambda)}}$ (1.8)

where $|\lambda|=\sum_{i}\lambda_{i}$.

At first sight it might be unclear why 1.1 is a natural generalization to
undertake, but we believe it to be in light of the wealth of other natural
$p$-adic random matrix ensembles which have found applications in number
theory and combinatorics. The iid Haar measure on nonsquare matrices was used
in [44] to model $p$-torsions of class groups of real quadratic number fields,
and the corresponding measure on cokernels was also related to Hall-Littlewood
polynomials in [39]. Measures on symmetric and antisymmetric $A$ have been
studied in [45] and [3], respectively, as models of sandpile groups of random
graphs and Tate-Shafarevich groups of elliptic curves, and have been related
to Hall-Littlewood polynomials in [20] and [21] respectively. Given the
utility of these models, it seems that any natural enough distribution on
$p$-adic random matrices is likely to model some class of random abelian
$p$-groups appearing in nature.

An expression for the probability in (1.7) was derived in [39], but in terms
of skew Hall-Littlewood polynomials rather than the explicit formula appearing
above. That result was suitable for asymptotics as the number of products went
to infinity, but seemed less adapted to studying the kinds of arithmetic
questions studied in the literature, such as the probability that the cokernel
is a cyclic group, for a finite number of products. The explicit nature of 1.1
appears more promising in this regard.

We note that one may give efficient sampling algorithms for the distribution
in 1.1 by interpreting the RHS of (1.7) in terms of steps of a certain Markov
chain, generalizing [19, Theorem 10]. It should also be possible to interpret
the $n\to\infty$ limit of (1.7) in terms of the appropriate notion of
automorphisms of a nested sequence of abelian $p$-groups, generalizing the
$k=1$ case (1.6); we have not attempted to address this question but hope it
will be taken up in the future.

###  1.6 Proof methods and skew Hall-Littlewood formulas.

A classical fact which follows from (1.1) is that when a geometric sequence
with common ratio $t$ is substituted in for the variables $x_{i}$, the so-
called _principal specialization_ , the Hall-Littlewood polynomial takes a
particularly simple form:

$P_{\lambda}(u,ut,\ldots,ut^{n-1};t)=u^{|\lambda|}t^{n(\lambda)}\frac{(t;t)_{n}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\lambda)}}.$
(1.9)

An explicit formula such as (1.1) is lacking for the skew Hall-Littlewood
polynomials, but their principal specializations still have a relatively
simple form. In the case of a specialization $u,ut,\ldots$ in infinitely many
variables this reads

$P_{\mu/\lambda}(u,ut,\ldots;t)=(t;t)_{\infty}u^{|\mu|-|\lambda|}t^{n(\mu/\lambda)}\prod_{x>0}\frac{(t^{1+\mu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}}{(t;t)_{m_{x}(\mu)}}$
(1.10)

for integer partitions $\mu,\lambda$, where $n(\mu/\lambda)$ generalizes
$n(\lambda)$ to skew diagrams–see 7. The formula (1.10) above is equivalent to
a formula for the modified Hall-Littlewood polynomials [27, Theorem 3.1], see
also [22, 43]. We give a different proof in Section 3 by degenerating formulas
for principally specialized skew higher spin Hall-Littlewood polynomials,
recently shown in [11], partially because we additionally need the result when
the geometric progression is finite, see 3.2. These formulas are the key
technical input in the proof of Section 1.2, and the proof of that result in
Section 4 relies on using these formulas to prove certain estimates (which are
not extremely difficult, once one has the formulas). They also imply
nontrivial variants of the _skew Cauchy identity_ in the special case of
principal specializations, which are derived and used in the algebraic
manipulations in the proof of Section 1.4. In 1.1 their role is even more
pronounced, as the result essentially follows directly from the formulas and
the existing result [39, Corollary 3.4] giving the probability in 1.1 in terms
of skew Hall-Littlewood polynomials.

###  1.7 Outline.

In Section 2 we set up notation concerning Hall-Littlewood polynomials, and in
Section 3 we prove formulas for principally specialized skew Hall-Littlewood
polynomials. These form the main tool for the classification of
$\partial\mathscr{G}_{t}$ in Section 4. In Section 5 we explain the setup of
$p$-adic random matrix theory, prove an augmented boundary result (5.4)
tailored to this situation, and use it to prove Section 1.3 and Section 1.4.
Finally, in Section 7 we prove 1.1. Finally, in Appendix A we prove a result
about Markov dynamics on $\partial\mathscr{G}_{t}$ which is motivated by a
parallel work [38].

### Acknowledgements

I am grateful to Alexei Borodin for many helpful conversations throughout the
project and detailed feedback on several drafts, Theo Assiotis and Alexander
Bufetov for comments and suggestions, Grigori Olshanski for discussions on
branching graphs, Vadim Gorin for feedback and the suggestion to recover
results of [1], Nathan Kaplan, Hoi Nguyen, and Melanie Matchett Wood for
discussions on cokernels of matrix products, and the anonymous referee for
helpful comments. This material is based on work partially supported by an NSF
Graduate Research Fellowship under grant #$1745302$, and by the NSF FRG grant
DMS-1664619.

##  2 Hall-Littlewood polynomials

In this section we give basic definitions of symmetric functions and Hall-
Littlewood polynomials. For a more detailed introduction to symmetric
functions see [29], and for Macdonald processes see [5].

###  2.1 Partitions, symmetric functions, and Hall-Littlewood polynomials.

We denote by $\mathbb{Y}$ the set of all integer partitions
$(\lambda_{1},\lambda_{2},\ldots)$, i.e. sequences of nonnegative integers
$\lambda_{1}\geq\lambda_{2}\geq\cdots$ which are eventually $0$. We call the
integers $\lambda_{i}$ the _parts_ of $\lambda$, set
$\lambda_{i}^{\prime}=\\#\\{j:\lambda_{j}\geq i\\}$, and write
$m_{i}(\lambda)=\\#\\{j:\lambda_{j}=i\\}=\lambda_{i}^{\prime}-\lambda_{i+1}^{\prime}$.
We write $\operatorname{len}(\lambda)$ for the number of nonzero parts, and
denote the set of partitions of length $\leq n$ by $\mathbb{Y}_{n}$. We write
$\mu\prec\lambda$ or $\lambda\succ\mu$ if
$\lambda_{1}\geq\mu_{1}\geq\lambda_{2}\geq\mu_{2}\geq\cdots$, and refer to
this condition as _interlacing_.

We denote by $\Lambda_{n}$ the ring $\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}$
of symmetric polynomials in $n$ variables $x_{1},\ldots,x_{n}$. It is a very
classical fact that the power sum symmetric polynomials
$p_{k}(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}x_{i}^{k},k=1,\ldots,n$, are
algebraically independent and algebraically generate $\Lambda_{n}$. For a
symmetric polynomial $f$, we will often write $f(\mathbf{x})$ for
$f(x_{1},\ldots,x_{n})$ when the number of variables is clear from context. We
will also use the shorthand
$\mathbf{x}^{\lambda}:=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots
x_{n}^{\lambda_{n}}$ for $\lambda\in\mathbb{Y}_{n}$.

One has a chain of maps

$\cdots\to\Lambda_{n+1}\to\Lambda_{n}\to\Lambda_{n-1}\to\cdots\to 0$

where the map $\Lambda_{n+1}\to\Lambda_{n}$ is given by setting $x_{n+1}$ to
$0$. In fact, writing $\Lambda_{n}^{(d)}$ for symmetric polynomials in $n$
variables of total degree $d$, one has

$\cdots\to\Lambda_{n+1}^{(d)}\to\Lambda_{n}^{(d)}\to\Lambda_{n-1}^{(d)}\to\cdots\to
0$

with the same maps. The inverse limit $\Lambda^{(d)}$ of these systems may be
viewed as symmetric polynomials of degree $d$ in infinitely many variables.
From the ring structure on each $\Lambda_{n}$ one gets a natural ring
structure on $\Lambda:=\bigoplus_{d\geq 0}\Lambda^{(d)}$, and we call this the
_ring of symmetric functions_. An equivalent definition is
$\Lambda:=\mathbb{C}[p_{1},p_{2},\ldots]$ where $p_{i}$ are indeterminates;
under the natural map $\Lambda\to\Lambda_{n}$ one has $p_{i}\mapsto
p_{i}(x_{1},\ldots,x_{n})$.

Each ring $\Lambda_{n}$ has a natural basis $\\{p_{\lambda}:\lambda_{1}\leq
n\\}$ where

$p_{\lambda}:=\prod_{i\geq 1}p_{\lambda_{i}}.$

Another natural basis, with the same index set, is given by the _Hall-
Littlewood polynomials_. Recall the $q$-Pochhammer symbol
$(a;q)_{n}:=\prod_{i=0}^{n-1}(1-aq^{i})$, and define

$v_{\lambda}(t)=\prod_{i\in\mathbb{Z}}\frac{(t;t)_{m_{i}(\lambda)}}{(1-t)^{m_{i}(\lambda)}}.$

######  Definition 1.

The Hall-Littlewood polynomial indexed by $\lambda\in\mathbb{Y}_{n}$ is

$P_{\lambda}(\mathbf{x};t)=\frac{1}{v_{\lambda}(t)}\sum_{\sigma\in
S_{n}}\sigma\left(\mathbf{x}^{\lambda}\prod_{1\leq i<j\leq
n}\frac{x_{i}-tx_{j}}{x_{i}-x_{j}}\right)$ (2.1)

where $\sigma$ acts by permuting the variables. We often drop the ‘$;t$’ when
clear from context.

It follows from the definition that

$P_{\lambda}(x_{1},\ldots,x_{n},0)=P_{\lambda}(x_{1},\ldots,x_{n}),$ (2.2)

hence for each $\lambda\in\mathbb{Y}$ there is a _Hall-Littlewood symmetric
function_ $P_{\lambda}\in\Lambda$.

In another direction, it is desirable to extend these definitions from
symmetric polynomials indexed by partitions to symmetric Laurent polynomials
indexed by _integer signatures_ with possibly negative parts. The set of
_integer signatures of length $n$_ is denoted

$\operatorname{\mathbb{GT}}_{n}:=\\{(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{Z}^{n}:\lambda_{1}\geq\ldots\geq\lambda_{n}\\}.$

The integers $\lambda_{n}$ are called _parts_ , as in the case of partitions.
We often identify $\lambda\in\mathbb{Y}_{n}$ with its image in
$\operatorname{\mathbb{GT}}_{n}$ by simply taking the first $n$ parts and
forgetting the zeroes which come after.
$\operatorname{\mathbb{GT}}_{n}^{>0}\subset\operatorname{\mathbb{GT}}_{n}$ is
the set of signatures with all parts positive, similarly for
$\operatorname{\mathbb{GT}}_{n}^{\geq 0}$. We set
$|\lambda|:=\sum_{i=1}^{n}\lambda_{i}$ and
$m_{k}(\lambda)=|\\{i:\lambda_{i}=k\\}|=\lambda_{k}^{\prime}-\lambda_{k+1}^{\prime}$
as with partitions. For $\lambda\in\operatorname{\mathbb{GT}}_{n}$ and
$\mu\in\operatorname{\mathbb{GT}}_{n-1}$, write $\mu\prec_{P}\lambda$ if
$\lambda_{i}\geq\mu_{i}$ and $\mu_{i}\geq\lambda_{i+1}$ for $1\leq i\leq n-1$.
For $\nu\in\operatorname{\mathbb{GT}}_{n}$ write $\nu\prec_{Q}\lambda$ if
$\lambda_{i}\geq\nu_{i}$ for $1\leq i\leq n$ and $\nu_{i}\geq\lambda_{i+1}$
for $1\leq i\leq n-1$. We write $c[k]$ for the signature $(c,\ldots,c)$ of
length $k$, and $()$ for the unique signature of length $0$. We often abuse
notation by writing $(\lambda,\mu)$ to refer to the tuple
$(\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots,\mu_{m})$ when
$\lambda\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{m}$.

1 extends from $\mathbb{Y}_{n}$ to $\operatorname{\mathbb{GT}}_{n}$ with no
other changes, and we will use the same notation $P_{\lambda}$ regardless of
whether $\lambda$ is a signature or a partition. It is also clear that

$P_{(\lambda_{1}+1,\ldots,\lambda_{n}+1)}(\mathbf{x};t)=x_{1}\cdots
x_{n}P_{\lambda}(\mathbf{x};t).$

######  Definition 2.

For $\lambda\in\operatorname{\mathbb{GT}}_{n}$, we define the dual Hall-
Littlewood polynomial by

$Q_{\lambda}(\mathbf{x};t)=\prod_{i\in\mathbb{Z}}(t;t)_{m_{i}(\lambda)}P_{\lambda}(\mathbf{x};t).$

We note that in the case where $\lambda\in\mathbb{Y}_{n}$ has some parts equal
to $0$, this normalization is _not_ the same as the standard one in e.g. [29],
though the two agree when $\lambda$ has all parts positive. We use this
nonstandard definition because parts equal to $0$ play no special role with
integer signatures, though they do in the usual setup with partitions. We will
see shortly that the classical results such as branching rules and the Cauchy
identity may be stated naturally in this setting.

Because the $P_{\lambda}$ form a basis for the vector space of symmetric
Laurent polynomials in $n$ variables, there exist symmetric Laurent
polynomials
$P_{\lambda/\mu}(x_{1},\ldots,x_{n-k};t)\in\Lambda_{n-k}[(x_{1}\cdots
x_{n-k})^{-1}]$ indexed by
$\lambda\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{k}$
which are defined by

$P_{\lambda}(x_{1},\ldots,x_{n};t)=\sum_{\mu\in\operatorname{\mathbb{GT}}_{k}}P_{\lambda/\mu}(x_{k+1},\ldots,x_{n};t)P_{\mu}(x_{1},\ldots,x_{k};t).$
(2.3)

We define the skew $Q$ functions in a slightly nonstandard way where the
lengths of both signatures are the same, in contrast to the skew $P$
functions; this is inspired by the higher spin Hall-Littlewood polynomials
introduced in [4].

######  Definition 3.

For $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{>0}$ and $k\geq 1$
arbitrary, define $Q_{\nu/\lambda}(x_{1},\ldots,x_{k};t)\in\lambda_{k}$ by

$Q_{(\nu,0[k])}(x_{1},\ldots,x_{n+k};t)=\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{>0}}Q_{\nu/\lambda}(x_{n+1},\ldots,x_{n+k};t)Q_{\lambda}(x_{1},\ldots,x_{k};t).$
(2.4)

In particular, $Q_{\lambda/(0[n])}(x_{1},\ldots,x_{n+k};t)$ agrees with
$Q_{(\lambda,0[k])}(x_{1},\ldots,x_{n+k};t)$ as defined earlier, and we will
use both interchangeably. Recall the two interlacing relations on signatures
$\succ_{P},\succ_{Q}$ defined above.

######  Definition 4.

For
$\mu\in\operatorname{\mathbb{GT}}_{n+1},\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$
with $\mu\succ_{P}\lambda,\nu\succ_{Q}\lambda$, let

$\psi_{\mu/\lambda}:=\prod_{\begin{subarray}{c}i\in\mathbb{Z}\\\
m_{i}(\lambda)=m_{i}(\mu)+1\end{subarray}}(1-t^{m_{i}(\lambda)})$

and

$\varphi_{\nu/\lambda}:=\prod_{\begin{subarray}{c}i\in\mathbb{Z}\\\
m_{i}(\nu)=m_{i}(\lambda)+1\end{subarray}}(1-t^{m_{i}(\nu)})$

The following branching rule is standard, but in this specific formulation
with signatures follows from [39, Lemma 2.1 and Proposition 2.8].

######  Lemma 2.1.

For
$\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{>0},\mu\in\operatorname{\mathbb{GT}}_{n-k}^{>0}$,
we have

$P_{\lambda/\mu}(x_{1},\ldots,x_{k})=\sum_{\mu=\lambda^{(1)}\prec_{P}\lambda^{(2)}\prec_{P}\cdots\prec_{P}\lambda^{(k)}=\lambda}\prod_{i=1}^{k-1}x_{i}^{|\lambda^{(i+1)}|-|\lambda^{(i)}|}\psi_{\lambda^{(i+1)}/\lambda^{(i)}}$
(2.5)

and

$Q_{\lambda/\nu}(x_{1},\ldots,x_{k})=\sum_{\nu=\lambda^{(1)}\prec_{Q}\lambda^{(2)}\prec_{Q}\cdots\prec_{Q}\lambda^{(k)}=\lambda}\prod_{i=1}^{k-1}x_{i}^{|\lambda^{(i+1)}|-|\lambda^{(i)}|}\varphi_{\lambda^{(i+1)}/\lambda^{(i)}}.$
(2.6)

The formulas from 2.1 may be used to define skew functions for general
signatures.

######  Definition 5.

For
$\lambda,\nu\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{n-k}$,
define $P_{\lambda/\mu}(x_{1},\ldots,x_{k})$ and
$Q_{\lambda/\nu}(x_{1},\ldots,x_{k})$ by the formulas (2.5) and (2.6)
respectively.

It follows from (2.2) and (2.3) that for
$\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0},\mu\in\operatorname{\mathbb{GT}}_{k}^{\geq 0}$,

$P_{\lambda/\mu}(x_{1},\ldots,x_{n-k})=P_{(\lambda,0)/(\mu,0)}(x_{1},\ldots,x_{n-k})=P_{(\lambda,0)/\mu}(x_{1},\ldots,x_{n-k},0).$
(2.7)

Therefore there exists a symmetric function
$P_{(\lambda,0,\ldots)/(\mu,0,\ldots)}\in\Lambda$ associated to the pair of
partitions $(\lambda,0,\ldots),(\mu,0,\ldots)\in\mathbb{Y}$, which maps to

$P_{\lambda/\mu}(x_{1},\ldots,x_{n-k})=P_{(\lambda,0)/(\mu,0)}(x_{1},\ldots,x_{n-k})$

under the map $\Lambda\to\Lambda_{n-k}$. With $Q$ the situation is slightly
more subtle: given either $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$ or
$\lambda,\nu\in\mathbb{Y}$ there exists an element
$Q_{\nu/\lambda}\in\Lambda$. If additionally
$\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$ then in fact
$Q_{\nu/\lambda}=Q_{(\nu,0)/(\lambda,0)}$. These properties can all be checked
from the above.

######  Remark 2.

It follows from 1 and 5 that for any $D\in\mathbb{Z}$,

$\displaystyle P_{\lambda+D[n]}(x_{1},\ldots,x_{n})$
$\displaystyle=(x_{1}\cdots x_{n})^{D}P_{\lambda}(x_{1},\ldots,x_{n})$
$\displaystyle P_{(\mu+D[m])/(\lambda+D[n])}(x_{1},\ldots,x_{m-n})$
$\displaystyle=(x_{1}\cdots x_{m-n})^{D}P_{\mu/\lambda}(x_{1},\ldots,x_{m-n})$

We note that $P_{\lambda/\mu}(x_{1},\ldots,x_{k})$ is in general a Laurent
polynomial, while $Q_{\lambda/\nu}(x_{1},\ldots,x_{k})$ is always a
polynomial.

Hall-Littlewood polynomials satisfy the _skew Cauchy identity_ , upon which
most probabilistic constructions rely. A formulation in terms of signatures is
given for the more general Macdonald polynomials in [39, Lemma 2.3], and the
below statement follows immediately by specializing that one.

######  Proposition 2.2.

Let
$\nu\in\operatorname{\mathbb{GT}}_{k},\mu\in\operatorname{\mathbb{GT}}_{n+k}$.
Then

$\sum_{\kappa\in\operatorname{\mathbb{GT}}_{n+k}}P_{\kappa/\nu}(x_{1},\ldots,x_{n};t)Q_{\kappa/\mu}(y_{1},\ldots,y_{m};t)\\\
=\prod_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq j\leq
m\end{subarray}}\frac{1-tx_{i}y_{j}}{1-x_{i}y_{j}}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{k}}Q_{\nu/\lambda}(y_{1},\ldots,y_{m};t)P_{\mu/\lambda}(x_{1},\ldots,x_{n};t).$
(2.8)

For later convenience we set

$\Pi(\mathbf{x};\mathbf{y}):=\prod_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq
j\leq
m\end{subarray}}\frac{1-tx_{i}y_{j}}{1-x_{i}y_{j}}=\exp\left(\sum_{\ell=1}^{\infty}\frac{1-t^{\ell}}{\ell}p_{\ell}(\mathbf{x})p_{\ell}(\mathbf{y})\right)$
(2.9)

(The second equality in (2.9) is not immediate but is shown in [29]).

There is also a more standard form of the Cauchy identity with integer
partitions rather than signatures, see [29]: For $\mu,\nu\in\mathbb{Y}$,

$\sum_{\kappa\in\mathbb{Y}}P_{\kappa/\nu}(x_{1},\ldots,x_{n};t)Q_{\kappa/\mu}(y_{1},\ldots,y_{m};t)\\\
=\prod_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq j\leq
m\end{subarray}}\frac{1-tx_{i}y_{j}}{1-x_{i}y_{j}}\sum_{\lambda\in\mathbb{Y}}Q_{\nu/\lambda}(y_{1},\ldots,y_{m};t)P_{\mu/\lambda}(x_{1},\ldots,x_{n};t).$
(2.10)

Hall-Littlewood polynomials/functions may be used to define Markovian dynamics
on $\operatorname{\mathbb{GT}}_{n}$. Given finite or infinite sequences
$\mathbf{a},\mathbf{b}$ of nonnegative real $a_{i},b_{j}$ with finite sums and
$a_{i}b_{j}<1$, the expressions

$\Pr(\lambda\to\nu):=Q_{\nu/\lambda}(\mathbf{a})\frac{P_{\nu}(\mathbf{b})}{P_{\lambda}(\mathbf{b})\Pi(\mathbf{a};\mathbf{b})}$
(2.11)

define transition probabilities by 2.2. The joint distribution of such
dynamics, run for $k$ steps with possibly distinct specializations
$\mathbf{a}^{(1)},\ldots,\mathbf{a}^{(k)}$ and started at $0[n]$, is a so-
called _ascending Hall-Littlewood process_. It is a measure on
$\operatorname{\mathbb{GT}}_{n}^{k}$ given by

$\Pr(\lambda^{(1)},\ldots,\lambda^{(k)})=\frac{P_{\lambda^{(k)}}(\mathbf{b})\prod_{i=1}^{k}Q_{\lambda^{(i)}/\lambda^{(i-1)}}(\mathbf{a}^{(i)})}{\Pi(\mathbf{b};\mathbf{a}^{(1)},\ldots,\mathbf{a}^{(k)})}$
(2.12)

where we take $\lambda^{(0)}=(0[n])$.

Finally, there are simple explicit formulas for the Hall-Littlewood
polynomials when a geometric progression $u,ut,\ldots,ut^{n-1}$ is substituted
for $x_{1},\ldots,x_{n}$–this is often referred as a _principal
specialization_. For $\lambda\in\operatorname{\mathbb{GT}}_{n}$ let

$n(\lambda):=\sum_{i=1}^{n}(i-1)\lambda_{i},$ (2.13)

and note that if additionally $\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}$ then

$n(\lambda)=\sum_{x\geq 1}\binom{\lambda_{x}^{\prime}}{2}.$

The following formula is standard. It may be easily derived from (2.1) by
noting that the summand is zero unless the permutation is the identity, and
evaluating this summand.

######  Proposition 2.3 (Principal specialization formula).

For $\lambda\in\operatorname{\mathbb{GT}}_{n}$,

$\displaystyle P_{\lambda}(u,ut,\ldots,ut^{n-1};t)$
$\displaystyle=u^{|\lambda|}t^{n(\lambda)}\frac{(t;t)_{n}}{\prod_{i\in\mathbb{Z}}(t;t)_{m_{i}(\lambda)}}.$

The proof sketched above offers no clear extension to the case of skew
polynomials, but in Section 3 we will derive such formulas using recent
results of [11].

##  3 Principally specialized skew Hall-Littlewood polynomials

In this section we prove (1.10) from the Introduction and its analogue for
$Q_{\nu/\lambda}$ in 3.3, as well as extensions when the geometric progression
is finite and the formulas are less simple in 3.2. Let us introduce a bare
minimum of background on higher spin Hall-Littlewood polynomials
$F_{\mu/\lambda},G_{\nu/\lambda}$, which generalize the usual Hall-Littlewood
polynomials $P,Q$ by the addition of an extra parameter $s$. We omit their
definition, which may be found in [4, 10], as we will only care about the case
$s=0$ when they reduce to slightly renormalized Hall-Littlewood polynomials.
When $s=0$, for $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{\geq
0},\mu\in\operatorname{\mathbb{GT}}_{n+k}^{\geq 0}$ one has

$F_{\mu/\lambda}(x_{1},\ldots,x_{k})\Big{|}_{s=0}=\prod_{i\geq
0}\frac{(t;t)_{m_{i}(\mu)}}{(t;t)_{m_{i}(\lambda)}}P_{\mu/\lambda}(x_{1},\ldots,x_{k})$
(3.1)

and

$G_{\nu/\lambda}(x_{1},\ldots,x_{k})\Big{|}_{s=0}=Q_{\nu/\lambda}(x_{1},\ldots,x_{k})$
(3.2)

by [4, §8.1]. Formulas for principally specialized skew $F$ and $G$ functions
were shown in [4], though we will state the version given later in [10]. We
apologize to the reader for giving a formula for an object which we have not
actually defined, but will immediately specialize to the Hall-Littlewood case,
so we hope no confusion arises. We need the following notation.

######  Definition 6.

The normalized terminating $q$-hypergeometric function is

$_{r+1}\bar{\phi}_{r}\left(\begin{matrix}t^{-n};a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{r}\end{matrix};t,z\right):=\sum_{k=0}^{n}z^{k}\frac{(t^{-n};t)_{k}}{(t;t)_{k}}\prod_{i=1}^{r}(a_{i};t)_{k}(b_{i}t^{k};t)_{n-k}$
(3.3)

for $n\in\mathbb{Z}_{\geq 0}$ and $|z|,|t|<1$.

######  Proposition 3.1 ([10, Proposition 5.5.1]).

Let $J\in\mathbb{Z}_{\geq 1},\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0},\mu\in\operatorname{\mathbb{GT}}_{n+J}^{\geq 0}$. Then

$F_{\mu/\lambda}(u,tu,\ldots,t^{J-1}u)=\prod_{x\in\mathbb{Z}_{\geq
0}}w_{u}^{(J)}(i_{1}(x),j_{1}(x);i_{2}(x),j_{2}(x)),$ (3.4)

where the product is over the unique collection of $n+J$ up-right paths on the
semi-infinite horizontal strip of height $1$ with paths entering from the
bottom at positions $\lambda_{i},1\leq i\leq n$, $J$ paths entering from the
left, and paths exiting from the top at positions $\mu_{i},1\leq i\leq n+J$,
see Figure 1. Here $i_{1}(x),j_{1}(x),i_{2}(x),j_{2}(x)$ are the number of
paths on the south, west, north and east edge of the vertex at position $x$ as
in Figure 2, and the weights in the product are given by

$w_{u}^{(J)}(i_{1},j_{1};i_{2},j_{2}):=\delta_{i_{1}+j_{1},i_{2}+j_{2}}\frac{(-1)^{i_{1}+j_{2}}t^{\frac{1}{2}i_{1}(i_{1}+2j_{1}-1)}s^{j_{2}-i_{1}}u^{i_{1}}(t;t)_{j_{1}}(us^{-1};t)_{j_{1}-i_{2}}}{(t;t)_{i_{1}}(t;t)_{j_{2}}(us;t)_{i_{1}+j_{1}}}\\\
\times\;_{4}\bar{\phi}_{3}\left(\begin{matrix}t^{-i_{1}};t^{-i_{2}},t^{J}su,tsu^{-1}\\\
s^{2},t^{1+j_{1}-i_{2}},t^{1+J-i_{1}-j_{1}}\end{matrix};t,t\right).$ (3.5)

Similarly, for $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$,
$G_{\nu/\lambda}(u,tu,\ldots,t^{J-1}u)$ is given by the product of the same
weights over the unique collection of $n$ up-right paths on the same strip
entering from the bottom at positions $\lambda_{i},1\leq i\leq n$ and exiting
from the top at positions $\nu_{i},1\leq i\leq n$.

0$\lambda_{n-1}=\lambda_{n}$$\lambda_{3}=\lambda_{2}$$\lambda_{1}$$\lambda_{4}$$\mu_{2}=\mu_{1}$$\mu_{3}$$\mu_{4}$$\mu_{n+J}$$\mu_{n+J-1}$

Figure 1:  The unique path collection corresponding to the function
$\mathbb{F}_{\mu/\lambda}(u,qu,\ldots,q^{J-1}u)$ with
$J=3,n=6,\lambda=(7,6,6,4,1,1),\mu=(8,8,6,4,2,2,2,1,0)$.
$i_{1}=2$$j_{1}=7$$i_{2}=5$$j_{2}=4$ Figure 2: Illustration of the notation
for edges, in the example $(i_{1},j_{1};i_{2},j_{2})=(2,7;5,4)$.

######  Remark 3.

To avoid confusion with [4, 10], we note that the parameter which we call $t$
for consistency with Hall-Littlewood notation is denoted by $q$ in these
references.

We now introduce some notation and specialize 3.1 to the Hall-Littlewood case
$s=0$.

######  Definition 7.

For $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$, we define

$n(\nu/\lambda):=\sum_{\begin{subarray}{c}1\leq i<j\leq
n\end{subarray}}\max(\nu_{j}-\lambda_{i},0)=\sum_{x\geq\lambda_{n}}\binom{\nu_{x+1}^{\prime}-\lambda_{x+1}^{\prime}}{2}.$

We additionally allow the case when $\lambda,\nu\in\mathbb{Y}$; the first
formula makes sense with the $\leq n$ removed, while for the second we simply
replace the sum over $x\geq\lambda_{n}$ by $x\geq 0$.

Note that $n(\nu/\lambda)$

  1. 1.

is translation-invariant, $n((\nu+D[n])/(\lambda+D[n]))=n(\nu/\lambda)$, and

  2. 2.

generalizes the standard definition of $n(\nu)$ in (2.13), namely when
$\nu\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$ then $n(\nu)=n(\nu/(0[n]))$.

One may also view $n(\nu/\lambda)$ as quantifying the failure of $\nu$ and
$\lambda$ to interlace; it is $0$ when $\nu,\lambda$ interlace, and increases
by $1$ when a part of $\lambda$ is moved past a part of $\nu$.

######  Proposition 3.2.

For $J\in\mathbb{Z}_{\geq 1},\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0},\mu\in\operatorname{\mathbb{GT}}_{n+J}^{\geq 0}$,

$\displaystyle
P_{\mu/\lambda}(u,\ldots,ut^{J-1})=(t;t)_{J}u^{|\mu|-|\lambda|}\prod_{x\geq
0}\frac{t^{m_{x}(\lambda)m_{x}(\mu)+\binom{\mu_{x+1}^{\prime}-\lambda_{x+1}^{\prime}}{2}}}{(t;t)_{m_{x}(\mu)}}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)},0\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right).$
(3.6)

For $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$,

$\displaystyle
Q_{\nu/\lambda}(u,\ldots,ut^{J-1})=u^{|\nu|-|\lambda|}t^{n(\nu/\lambda)}\prod_{x\in\mathbb{Z}}\frac{t^{m_{x}(\lambda)m_{x}(\nu)}}{(t;t)_{m_{x}(\lambda)}}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\nu)},0\\\
t^{1+\nu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\nu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right).$
(3.7)

###### Proof.

We begin with (3.6). In this case we may apply 3.1 to compute

$\text{LHS\eqref{eq:finite_skew_formula}}=F_{\mu/\lambda}(u,\ldots,ut^{J-1})\Big{|}_{s=0}\prod_{i\geq
0}\frac{(t;t)_{m_{i}(\lambda)}}{(t;t)_{m_{i}(\mu)}}.$ (3.8)

When $s\to 0$, the factor $s^{j_{2}-i_{1}}(us^{-1};t)_{j_{1}-i_{2}}$ in (3.5)
converges to $(-u)^{j_{1}-i_{2}}t^{\binom{j_{1}-i_{2}}{2}}$ (using that
$j_{2}-i_{1}=j_{1}-i_{2}$). The sign cancels with the sign in (3.5), and the
power of $u$ combines with the $u^{i_{1}}$ in (3.5) to give $u^{j_{2}}$, so
(3.5) becomes

$w_{t^{n}}^{(J)}(i_{1},j_{1};i_{2},j_{2})=\delta_{i_{1}+j_{1},i_{2}+j_{2}}u^{j_{2}}\frac{t^{\frac{1}{2}i_{1}(i_{1}+2j_{1}-1)+\binom{j_{1}-i_{2}}{2}}(t;t)_{j_{1}}}{(t;t)_{i_{1}}(t;t)_{j_{2}}}\;_{4}\bar{\phi}_{3}\left(\begin{matrix}t^{-i_{1}};t^{-i_{2}},0,0\\\
0,t^{1+j_{1}-i_{2}},t^{1+J-i_{1}-j_{1}}\end{matrix};t,t\right).$ (3.9)

In the product (3.4) when the weights are specialized to (3.9), some of the
factors simplify, as

$\prod_{x\geq
0}\frac{(t;t)_{j_{1}}}{(t;t)_{i_{1}}(t;t)_{j_{2}}}=\frac{(t;t)_{J}}{\prod_{x\in\mathbb{Z}}(t;t)_{i_{1}(x)}}$
(3.10)

because the $\frac{(t;t)_{j_{1}(x)}}{(t;t)_{j_{2}(x)}}$ factor cancels except
for a $(t;t)_{J}$ from the paths incoming from the left. Hence

$\prod_{x\geq 0}w_{u}^{(J)}(i_{1},j_{1};i_{2},j_{2})=(t;t)_{J}\prod_{x\geq
0}u^{j_{2}}\frac{t^{\frac{1}{2}i_{1}(i_{1}+2j_{1}-1)+\binom{j_{1}-i_{2}}{2}}}{(t;t)_{m_{x}(\lambda)}}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-i_{1}};t^{-i_{2}},0\\\
t^{1+j_{1}-i_{2}},t^{1+J-i_{1}-j_{1}}\end{matrix};t,t\right).$ (3.11)

Using that $j_{2}=i_{1}+j_{1}-i_{2}$ simplifies the exponent of $t$ in (3.11)
to

$\frac{1}{2}i_{1}(i_{1}+2j_{1}-1)+\binom{j_{1}-i_{2}}{2}=\binom{j_{2}}{2}+i_{1}i_{2}.$

To convert to the form in terms of partitions, we record the following
translations between the $i$’s and $j$’s and the usual conjugate partition
notation:

$\displaystyle\begin{split}i_{1}(x)&=\lambda_{x}^{\prime}-\lambda_{x+1}^{\prime}=m_{x}(\lambda),\\\
j_{1}(x)&=\mu_{x}^{\prime}-\lambda_{x}^{\prime},\\\
i_{2}(x)&=\mu_{x}^{\prime}-\mu_{x+1}^{\prime}=m_{x}(\mu),\\\
j_{2}(x)&=\mu_{x+1}^{\prime}-\lambda_{x+1}^{\prime}.\end{split}$ (3.12)

Translating (3.11) into partition notation and multiplying by the
$\prod_{i\geq 0}\frac{(t;t)_{m_{i}(\lambda)}}{(t;t)_{m_{i}(\mu)}}$ factor of
(3.8) yields (3.6).

To prove (3.7) we first note that both sides of (3.7) are translation-
invariant, so without loss of generality we may take
$\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{>0}$. We then likewise appeal
to 3.1 and either make the same argument as above or deduce it from (3.6) by
considering $F_{(\nu,0[J])/\lambda}(u,\ldots,ut^{J-1})$ for
$\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}^{>0}$ and using the relation
between $P$ and $Q$ polynomials. Since $\lambda,\nu$ are of the same length we
have

$\prod_{x\geq 0}t^{\binom{j_{2}(x)}{2}}=t^{n(\nu/\lambda)}$

by (3.12). Finally, note that the product can be extended from $x\geq 0$ to
$x\in\mathbb{Z}$, which in this translation-invariant setting is more
aesthetically appealing. ∎

######  Remark 4.

While it follows from the branching rule that for nonnegative signatures
$\mu,\lambda$ of appropriate lengths,

$P_{(\mu,0)/(\lambda,0)}(u,\ldots,ut^{J-1})=P_{\mu/\lambda}(u,\ldots,ut^{J-1}),$

see (2.7), this relation is not readily apparent from (3.6). The only term on
the RHS of (3.6) which a priori might differ after padding $\lambda,\mu$ with
zeros is the $x=0$ term of the product. It may be checked that this term is in
fact unchanged by padding with zeros, but this is not immediately obvious from
the formula as written.

The next result takes the $J\to\infty$ limit of 3.2. Recall from Section 2
that if $\mu,\lambda\in\mathbb{Y}$, $P_{\mu/\lambda}\in\Lambda$ is a
polynomial in the power sums $p_{1},p_{2},\ldots$. Hence given any infinite
sequence of complex numbers $a_{1},a_{2},\ldots$ such that the sums
$p_{k}(a_{1},a_{2},\ldots)$ converge (it suffices for this to hold for
$p_{1}$), we may define $P_{\mu/\lambda}(a_{1},a_{2},\ldots)\in\mathbb{C}$.
This is how $P_{\mu/\lambda}(u,ut,\ldots)$ is to be interpreted below, and
similarly for $Q_{\nu/\lambda}(u,ut,\ldots)$.

######  Theorem 3.3.

For $\mu,\lambda\in\mathbb{Y}$, we have

$\displaystyle\begin{split}P_{\mu/\lambda}(u,ut,\ldots)&=u^{|\mu|-|\lambda|}t^{n(\mu/\lambda)}\prod_{x\geq
1}\frac{(t^{1+\mu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}}{(t;t)_{m_{x}(\mu)}}.\end{split}$
(3.13)

For $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$,

$\displaystyle
Q_{\nu/\lambda}(u,ut,\ldots)=u^{|\nu|-|\lambda|}t^{n(\nu/\lambda)}\prod_{x\in\mathbb{Z}}\frac{(t^{1+\nu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}}{(t;t)_{m_{x}(\lambda)}}$
(3.14)

###### Proof.

For $n\geq\operatorname{len}(\lambda),n+J\geq\operatorname{len}(\mu)$, we may
identify $\mu,\lambda\in\mathbb{Y}$ with nonnegative signatures
$\mu(n+J)\in\operatorname{\mathbb{GT}}_{n+J}^{\geq
0},\lambda(n)\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$ given by truncating.
Hence to compute

$P_{\mu/\lambda}(u,ut,\ldots)$

it suffices to take $J\to\infty$ in (3.6). The polynomial
$P_{\mu(n+J)/\lambda(n)}$ is independent of $n$ for all $n$ sufficiently
large, see (2.7), so we will fix $n$ and will abuse notation below and write
$\lambda$ for $\lambda(n)$. We first pull the $1/(t;t)_{m_{x}(\mu)}$ out of
the product, and note that $m_{0}(\mu(n+J))\to(t;t)_{\infty}$ as $J\to\infty$,
cancelling the $(t;t)_{J}$ term of (3.6). We write the remaining term inside
the product in (3.6) as

$\left(t^{m_{x}(\lambda)m_{x}(\mu(n+J))}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu(n+J))},0\\\
t^{1+\mu(n+J)_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu(n+J)_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right)\right).$
(3.15)

To show (3.13) it suffices to show that for $x>0$,

$\lim_{J\to\infty}t^{m_{x}(\lambda)m_{x}(\mu(n+J))}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu(n+J))},0\\\
t^{1+\mu(n+J)_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu(n+J)_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right)=(t^{1+\mu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}.$
(3.16)

and for $x=0$,

$\lim_{J\to\infty}t^{m_{x}(\lambda)m_{x}(\mu(n+J))}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu(n+J))},0\\\
t^{1+\mu(n+J)_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu(n+J)_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right)=1.$
(3.17)

We begin with (3.13). Then
$1+J-\mu(n+J)_{x}^{\prime}+\lambda_{x+1}^{\prime}\to\infty$ and all other
arguments in the $q$-hypergeometric function remain the same, so the LHS of
(3.16) is

$\;_{2}\bar{\phi}_{1}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)}\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}}\end{matrix};t,t\right)=\sum_{\ell=0}^{m_{x}(\lambda)}t^{\ell}\frac{(t^{-m_{x}(\lambda)};t)_{\ell}}{(t;t)_{\ell}}(t^{-m_{x}(\mu)};t)_{\ell}(t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}+\ell};t)_{m_{x}(\lambda)-\ell}.$
(3.18)

To apply known identities, we reexpress the above in terms of the more
standard terminating $q$-hypergeometric series $\;{}_{2}\phi_{1}$ as

$(t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}\sum_{\ell=0}^{m_{x}(\lambda)}t^{\ell}\frac{(t^{-m_{x}(\lambda)};t)_{\ell}(t^{-m_{x}(\mu)};t)_{\ell}}{(t;t)_{\ell}(t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}};t)_{\ell}}\\\
=(t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}\;{}_{2}\phi_{1}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)}\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}}\end{matrix};t,t\right).$ (3.19)

By a special case of the $q$-Gauss identity, see e.g. [28, Exercise 3.17],

$_{2}\phi_{1}\left(\begin{matrix}t^{-n};b\\\
c\end{matrix};t,t\right)=\frac{(c/b;t)_{n}}{(c;t)_{n}}b^{n}.$ (3.20)

Applying (3.20) with
$b=t^{-m_{x}(\mu)},c=t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}}$ to (3.19)
yields

$\;_{2}\bar{\phi}_{1}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)}\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}}\end{matrix};t,t\right)=(t^{1+\mu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}t^{-m_{x}(\lambda)m_{x}(\mu)},$
(3.21)

which shows (3.16).

We now show (3.17), so let $x=0$. Then $\mu_{x}(n+J)^{\prime}=n+J$, so the
arguments of the $q$-hypergeometric function in (3.16) are independent of $J$
except for $t^{-m_{0}(\mu(n+J))}$. In the sum

$t^{m_{x}(\lambda)m_{x}(\mu(n+J))}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu(n+J))},0\\\
t^{1+\mu(n+J)_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu(n+J)_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right)=t^{m_{x}(\lambda)m_{x}(\mu(n+J))}\\\
\times\sum_{k=0}^{m_{x}(\lambda)}t^{k}\frac{(t^{-m_{x}(\lambda)};t)_{k}}{(t;t)_{k}}(t^{-m_{x}(\mu(n+J))};t)_{k}(t^{k+1+\mu(n+J)_{x+1}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)-k}(t^{k+1-n+\lambda_{x+1}^{\prime}};t)_{m_{x}(\lambda)-k},$

the dominant term as $J\to\infty$ is the $k=m_{x}(\lambda)$ term, and its
limit when normalized by $t^{m_{x}(\lambda)m_{x}(\mu(n+J))}$ is $1$. This
shows (3.17).

The proof of (3.14) using (3.7) is exactly analogous except that only (3.16)
is needed because there are only $n$ paths. ∎

##  4 The $t$-deformed Gelfand-Tsetlin graph and its boundary

Let $t\in(0,1)$ for the remainder of the section. In this section we introduce
the Hall-Littlewood Gelfand-Tsetlin graph and the notion of its boundary, the
set of extreme coherent systems. The main result stated earlier, Section 1.2,
is that the boundary is naturally in bijection with the set
$\operatorname{\mathbb{GT}}_{\infty}$ of infinite signatures. We will break it
into three parts: 4.2 gives an explicit coherent system of measures
$(M_{n}^{\mu})_{n\geq 1}$ for each
$\mu\in\operatorname{\mathbb{GT}}_{\infty}$, 4.7 tells that every extreme
coherent system must be one of these, and 4.10 tells that each system
$(M_{n}^{\mu})_{n\geq 1}$ is extreme.

The general structure of the proof of Section 1.2, via the so-called Vershik-
Kerov ergodic method, is similar to e.g. [34, Theorem 6.2] or [15]. A good
general reference for (unweighted) graded graphs, with references to research
articles, is [9, Chapter 7].

###  4.1 Classifying the boundary.

######  Definition 8.

$\mathscr{G}_{t}$ is the weighted, graded graph with vertices

$\bigsqcup_{n\geq 1}\operatorname{\mathbb{GT}}_{n}$

partitioned into _levels_ indexed by $\mathbb{Z}_{\geq 1}$. The only edges of
$\mathscr{G}_{t}$ are between vertices on levels differing by $1$. Between
every
$\lambda\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{n+1}$
there is a weighted edge with weight

$L_{n}^{n+1}(\mu,\lambda):=P_{\mu/\lambda}(t^{n})\frac{P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,\ldots,t^{n})},$

and these weights are called _cotransition probabilities_ or _(stochastic)
links_. We use $L^{n+1}_{n}$ to denote the (infinite)
$\operatorname{\mathbb{GT}}_{n+1}\times\operatorname{\mathbb{GT}}_{n}$ matrix
with these weights.

Note $L^{n+1}_{n}$ is a stochastic matrix by the branching rule. More
generally, for $m\in\mathbb{Z}_{\geq 1}\cup\\{\infty\\}$, $1\leq n<m$, and
$\mu\in\operatorname{\mathbb{GT}}_{m},\lambda\in\operatorname{\mathbb{GT}}_{n}$
we let

$L^{m}_{n}(\mu,\lambda):=P_{\mu/\lambda}(t^{n},\ldots,t^{m-1})\frac{P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,\ldots,t^{m-1})}.$
(4.1)

When $m$ is finite one has $L^{m}_{n}=L^{n+1}_{n}L^{n+2}_{n}\cdots
L^{m}_{m-1}$, where the product is just the usual matrix product.

######  Remark 5.

The cotransition probabilities define (deterministic) maps
$\mathcal{M}(\operatorname{\mathbb{GT}}_{m})\to\mathcal{M}(\operatorname{\mathbb{GT}}_{n})$,
where here and below we use $\mathcal{M}$ to denote the space of Borel
probability measures, in this case with respect to the discrete topology on
the set of signatures.

######  Remark 6.

2 implies translation-invariance

$L^{m}_{n}(\mu,\lambda)=L^{m}_{n}(\mu+D[m],\lambda+D[n])$ (4.2)

of the cotransition probabilities.

The cotransition probabilities have explicit formulas courtesy of the results
of Section 3, which will be useful in the proofs of 4.5 and 4.7 later. For
$\lambda\in\operatorname{\mathbb{GT}}_{n}$, we let

$\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}=\frac{(t;t)_{n}}{\prod_{i\in\mathbb{Z}}(t;t)_{m_{i}(\lambda)}}$

(the $t$-deformed multinomial coefficient).

######  Corollary 4.1.

For
$\mu\in\operatorname{\mathbb{GT}}_{n+J},\lambda\in\operatorname{\mathbb{GT}}_{n}$,

$L^{n+J}_{n}(\mu,\lambda)=\frac{1}{\begin{bmatrix}n+J\\\
J\end{bmatrix}_{t}}\prod_{x\in\mathbb{Z}}\frac{t^{(n-\lambda_{x}^{\prime})(\mu_{x}^{\prime}-\lambda_{x}^{\prime})+m_{x}(\lambda)m_{x}(\mu)}}{(t;t)_{m_{x}(\lambda)}}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)},0\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right).$
(4.3)

###### Proof.

By the translation-invariance of 6, it suffices to prove the case when
$\mu,\lambda$ are nonnegative signatures. We combine the formula of 3.2 for
$P_{\mu/\lambda}(t^{n},\ldots,t^{n+J-1})$ with the one from 2.3 for the
principally specialized non-skew Hall-Littlewood polynomial. By the latter,

$\frac{P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,\ldots,t^{n+J-1})}=\frac{(t;t)_{n}}{(t;t)_{n+J}}t^{n(\lambda)-n(\mu)}\prod_{i\geq
0}\frac{(t;t)_{m_{i}(\mu)}}{(t;t)_{m_{i}(\lambda)}}.$ (4.4)

Note also that by the definition of $n(\lambda)$,

$t^{n(\lambda)-n(\mu)}=\prod_{x\geq
0}t^{\binom{\lambda_{x+1}^{\prime}}{2}-\binom{\mu_{x+1}^{\prime}}{2}},$ (4.5)

so by the identity

$\binom{a+b}{2}-\binom{a}{2}-\binom{b}{2}=ab$

we have

$t^{n(\lambda)-n(\mu)}\prod_{x\geq
0}t^{\binom{\mu_{x+1}^{\prime}-\lambda_{x+1}^{\prime}}{2}}=\prod_{x\geq
0}t^{-\lambda_{x+1}^{\prime}(\mu_{x+1}^{\prime}-\lambda_{x+1}^{\prime})}.$
(4.6)

Simplifying the product of (3.6) with (4.4) by the above manipulations yields

$\displaystyle\begin{split}L^{n+J}_{n}(\mu,\lambda)=\frac{1}{\begin{bmatrix}n+J\\\
J\end{bmatrix}_{t}}\prod_{x\geq
0}\frac{t^{(n-\lambda_{x+1}^{\prime})(\mu_{x+1}^{\prime}-\lambda_{x+1}^{\prime})+m_{x}(\lambda)m_{x}(\mu)}}{(t;t)_{m_{x}(\lambda)}}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\mu)},0\\\
t^{1+\mu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\mu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right).\end{split}$

The product may be extended to all $x\in\mathbb{Z}$ since all other terms are
$1$, at which point it is manifestly translation-invariant, which yields the
result for arbitrary signatures. ∎

######  Definition 9.

A sequence $(M_{n})_{n\geq 1}$ of probability measures on
$\operatorname{\mathbb{GT}}_{1},\operatorname{\mathbb{GT}}_{2},\ldots$ is
_coherent_ if

$\sum_{\mu\in\operatorname{\mathbb{GT}}_{n+1}}M_{n+1}(\mu)L_{n}^{n+1}(\mu,\lambda)=M_{n}(\lambda)$

for each $n\geq 1$ and $\lambda\in\operatorname{\mathbb{GT}}_{n}$.

######  Definition 10.

A coherent system of measures $(M_{n})_{n\geq 1}$ is _extreme_ if there do not
exist coherent systems $(M_{n}^{\prime})_{n\geq
1},(M_{n}^{\prime\prime})_{n\geq 1}$ different from $(M_{n})_{n\geq 1}$ and
$s\in(0,1)$ such that $M_{n}=sM_{n}^{\prime}+(1-s)M_{n}^{\prime\prime}$ for
each $n$. The set of extreme coherent systems of measures on a weighted,
graded graph is called its _boundary_ , and denoted in our case by
$\partial\mathscr{G}_{t}$.

In the previous section we considered both signatures (of finite length), and
integer partitions, which have infinite length but stabilize to $0$. To
describe points on the boundary $\partial\mathscr{G}_{t}$ in this section, it
turns out that it will be necessary to introduce signatures of infinite length
which are not partitions.

######  Definition 11.

We denote the set of infinite signatures by

$\operatorname{\mathbb{GT}}_{\infty}:=\\{(\mu_{1},\mu_{2},\ldots)\in\mathbb{Z}^{\infty}:\mu_{1}\geq\mu_{2}\geq\ldots\\}.$

We refer to the $\mu_{i}$ as _parts_ just as with partitions, and define
$\mu_{i}^{\prime}$ and $m_{i}(\mu)$ the exact same way, though we must allow
them to be equal to $\infty$.

A distinguished subset of $\operatorname{\mathbb{GT}}_{\infty}$ is
$\mathbb{Y}$, the set of partitions. Translating by any $D\in\mathbb{Z}$
yields

$\mathbb{Y}+D=\\{\mu\in\operatorname{\mathbb{GT}}_{\infty}:\mu_{i}=D\text{ for
all but finitely many }i\\}.$

However, $\operatorname{\mathbb{GT}}_{\infty}$ also contains infinite
signatures with parts not bounded below, the set of which we denote by

$\operatorname{\mathbb{GT}}_{\infty}^{unstable}:=\\{\mu\in\operatorname{\mathbb{GT}}_{\infty}:\lim_{i\to\infty}\mu_{i}=-\infty\\}.$

It is clear that

$\operatorname{\mathbb{GT}}_{\infty}=\operatorname{\mathbb{GT}}_{\infty}^{unstable}\sqcup\bigsqcup_{D\in\mathbb{Z}}(\mathbb{Y}+D)$

and we will use this decomposition repeatedly in what follows. To treat the
unbounded signatures we will approximate by signatures in $\mathbb{Y}+D$,
which are no more complicated than partitions, and to this end we introduce
the following notation.

######  Definition 12.

For $\lambda\in\operatorname{\mathbb{GT}}_{\infty}^{unstable}$ and
$D\in\mathbb{Z}$, we let

$\lambda^{(D)}=(\lambda_{1},\ldots,\lambda_{k},D,D,\ldots)\in\mathbb{Y}+D$

where $k$ is the largest index such that $\lambda_{k}>D$.

The first step to proving Section 1.2 is, for each element of
$\operatorname{\mathbb{GT}}_{\infty}$, an explicit formula for a corresponding
coherent system of measures on $\mathscr{G}_{t}$; we will later show that
these are exactly the boundary points.

######  Proposition 4.2.

For each $\mu\in\operatorname{\mathbb{GT}}_{\infty}$, there exists a coherent
system of measures $(M_{n}^{\mu})_{n\geq 1}$ on $\mathscr{G}_{t}$, given
explicitly by

$M_{n}^{\mu}(\lambda):=\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}(t^{1+\mu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}.$
(4.7)

for $\lambda\in\operatorname{\mathbb{GT}}_{n}$.

Before proving 4.2 we will calculate explicit formulas for the links
$L^{m}_{n}$ in 3.2, which are a corollary to the formula for principally
specialized skew functions in 3.3.

######  Corollary 4.3.

Let $n\geq 1$. If $\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0},\mu\in\mathbb{Y}$, then

$\frac{P_{\mu/\lambda}(t^{n},t^{n+1},\ldots)P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,t,\ldots)}=\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}_{>0}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}(t^{1+\mu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}.$
(4.8)

Furthermore, if instead
$\lambda\in\operatorname{\mathbb{GT}}_{n},\mu\in\operatorname{\mathbb{GT}}_{\infty}^{unstable}$,
then

$\frac{P_{(\mu^{(D)}-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}$
(4.9)

increases monotonically as $D\to-\infty$, and stabilizes to

$\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}(t^{1+\mu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}$
(4.10)

for all $D<\lambda_{n}$.

###### Proof.

(4.8) follows from 3.3 and 2.3 by the same proof as that of 4.1, so let us
show the monotonicity and stabilization statement. Substituting (4.9) into
(4.8) and changing variables $x\mapsto x+D$ in the product yields

$\frac{P_{(\mu^{(D)}-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}=\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}_{>D}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}(t^{1+\mu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}.$

The factors in the product are all in $[0,1]$ and are equal to $1$ when
$x\leq\lambda_{n}$, and since the product is over $x\in\mathbb{Z}_{>D}$ this
completes the proof. ∎

######  Remark 7.

Given the translation-invariance of the links $L^{m}_{n}$ noted in 6, when
$\mu\in\mathbb{Y}+D$ it is natural to view the expression

$\frac{P_{(\mu-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu-D[\infty])}(1,t,\ldots)}$

as simply

$\frac{P_{\mu/\lambda}(t^{n},t^{n+1},\ldots)P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,t,\ldots)},$

even though in our setup the expressions
$P_{\mu/\lambda}(t^{n},t^{n+1},\ldots)$ and $P_{\mu}(1,t,\ldots)$ are not
well-defined when $\mu$ is not in $\mathbb{Y}$. Hence in view of 3.3 it is
natural to view the coherent systems $(M^{\mu}_{n})_{n\geq 1}$ of Section 1.2
as being given by links ‘at infinity’

$M^{\mu}_{n}(\lambda)\text{``}=\text{''}L^{\infty}_{n}(\mu,\lambda)=\frac{P_{\mu/\lambda}(t^{n},t^{n+1},\ldots)P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,t,\ldots)}$

for general $\mu\in\operatorname{\mathbb{GT}}_{\infty}$, though we must take a
slightly roundabout path to make rigorous sense of the RHS. Many of the proofs
below follow the same pattern of proving a result for $\mu\in\mathbb{Y}$ by
usual symmetric functions machinery, appealing to translation-invariance for
$\mu\in\mathbb{Y}+D$, and then approximating
$\mu\in\operatorname{\mathbb{GT}}_{\infty}^{unstable}$ by elements
$\mu^{(D)}\in\mathbb{Y}+D$ and using 4.3 to apply the monotone convergence
theorem. We note also that the formula (4.7) is clearly translation-invariant.

###### Proof of 4.2.

We first show $M_{n}^{\mu}$ is indeed a probability measure. Clearly it is a
nonnegative function on $\operatorname{\mathbb{GT}}_{n}$, but we must show it
sums to $1$. When $\mu\in\mathbb{Y}$ this is by 4.3 and the definition of skew
HL polynomials, and the case $\mu\in\mathbb{Y}+D$ reduces to this one. Hence
it remains to show that for
$\mu\in\operatorname{\mathbb{GT}}_{\infty}^{unstable}$,

$\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}}\lim_{D\to-\infty}\frac{P_{(\mu^{(D)}-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}=1.$
(4.11)

By 4.3, the functions

$\frac{P_{(\mu^{(D)}-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}=\begin{bmatrix}n\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}_{>D}}t^{(\mu^{\prime}_{x}-\lambda^{\prime}_{x})(n-\lambda^{\prime}_{x})}(t^{1+\mu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}$

converge to the summand in (4.11) from below as $D\to-\infty$. Hence (4.11)
follows by the monotone convergence theorem.

For $\mu\in\mathbb{Y}+D$ for some $D$, coherency again follows from the
definition of skew functions and the first part of 4.3. For
$\mu\in\operatorname{\mathbb{GT}}_{\infty}^{unstable}$ we must show

$\sum_{\kappa\in\operatorname{\mathbb{GT}}_{n+1}}\lim_{D\to-\infty}\frac{P_{(\mu^{(D)}-D[\infty])/(\kappa-D[n+1])}(t^{n+1},\ldots)P_{(\kappa-D[n+1])}(1,\ldots,t^{n})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}\frac{P_{\kappa/\lambda}(t^{n})P_{\lambda}(1,\ldots,t^{n-1})}{P_{\kappa}(1,\ldots,t^{n})}\\\
=\lim_{D\to-\infty}\frac{P_{(\mu^{(D)}-D[\infty])/(\lambda-D[n])}(t^{n},t^{n+1},\ldots)P_{(\lambda-D[n])}(1,\ldots,t^{n-1})}{P_{(\mu^{(D)}-D[\infty])}(1,t,\ldots)}.$
(4.12)

Again the monotone convergence theorem allows us to interchange the limit and
sum. The result then follows by translation invariance of the links (4.2) and
the definition of skew HL polynomials. ∎

It remains to show that the coherent systems identified in 4.2 are extreme and
that all extreme coherent systems are of this form. Just from the definition,
an arbitrary extreme coherent system is an elusive object. Luckily, the
general results of the Vershik-Kerov ergodic method guarantee that extreme
coherent systems can be obtained through limits of cotransition probabilities
for certain _regular sequences_ of signatures, which are much more concrete.

######  Definition 13.

A sequence $(\mu(n))_{n\geq 1}$ with $\mu(n)\in\operatorname{\mathbb{GT}}_{n}$
is _regular_ if for every $k\in\mathbb{Z}_{\geq 1}$ and
$\lambda\in\operatorname{\mathbb{GT}}_{k}$, the limit

$M_{k}(\lambda):=\lim_{n\to\infty}L^{n}_{k}(\mu(n),\lambda)$

exists and $M_{k}$ is a probability measure.

######  Proposition 4.4.

For any extreme coherent system $(M_{k})_{k\geq 1}\in\partial G$ there exists
a regular sequence $(\mu(n))_{n\geq 1}$ such that

$M_{k}(\cdot)=\lim_{n\to\infty}L_{k}^{n}(\mu(n),\cdot).$

###### Proof.

Follows from [32, Theorem 6.1]. ∎

The space of extreme coherent systems obtained from regular sequences as in
4.4 is sometimes referred to as the _Martin boundary_. It naturally includes
into the boundary, and 4.4 says that in this setup they are in fact equal.

######  Lemma 4.5.

Let $(\mu(n))_{n\geq 1}$ be a sequence with
$\mu(n)\in\operatorname{\mathbb{GT}}_{n}$, such that

$\lim_{n\to\infty}\mu(n)_{i}=:\mu_{i}$

exists and is finite for every $i$. Then $(\mu(n))_{n\geq 1}$ is regular and
the corresponding coherent family is $(M_{n}^{\mu})_{n\geq 1}$, where
$\mu=(\mu_{1},\mu_{2},\ldots)\in\operatorname{\mathbb{GT}}_{\infty}$.

###### Proof.

Let $(\mu(n))_{n\geq 1}$ satisfy the hypothesis. We must show for arbitrary
$k,\lambda\in\operatorname{\mathbb{GT}}_{k}$ that

$\lim_{n\to\infty}L_{k}^{n}(\mu(n),\lambda)=M^{\mu}_{k}(\lambda).$ (4.13)

It is easy to see from the explicit formula in 4.1 that
$L_{k}^{n}(\mu(n),\lambda)$ depends only on the parts of $\mu(n)$ which are
$\geq\lambda_{k}$. For any fixed $x$, it is easy to see that
$\mu(n)_{x}^{\prime}\to\mu_{x}^{\prime}$. In fact, for all sufficiently large
$n$, it must be true that $\mu(n)_{x}^{\prime}=\mu_{x}^{\prime}$ for all
$x\geq\lambda_{k}$ such that $\mu_{x}^{\prime}$ is finite. Hence for all
sufficiently large $n$ the product in (4.3) only has nontrivial terms when
$\lambda_{k}\leq x\leq\mu_{1}$, so it suffices to show that each term
converges. This follows by the exact same argument as the proof of 3.3, again
with two cases based on whether $\mu(n)_{x}^{\prime}$ stabilizes or
$\mu(n)_{x}^{\prime}\to\infty$. ∎

######  Lemma 4.6.

Let $1\leq k<n$ be integers and $\lambda\in\operatorname{\mathbb{GT}}_{k}$.

  1. 1.

If $\mu\in\operatorname{\mathbb{GT}}_{n}$ is such that
$\lambda^{\prime}_{x}>\mu^{\prime}_{x}$ for some $x$, then
$L_{k}^{n}(\mu,\lambda)=0$.

  2. 2.

If $\mu\in\operatorname{\mathbb{GT}}_{\infty}$ is such that
$\lambda^{\prime}_{x}>\mu^{\prime}_{x}$ for some $x$, then
$M_{k}^{\mu}(\lambda)=0$.

###### Proof.

If $\mu\in\mathbb{Y},\lambda\in\mathbb{Y}$ and
$\lambda^{\prime}_{x}>\mu^{\prime}_{x}$ for any $x$, it follows from the
upper-triangularity of the branching rule [29, Chapter III, (5.5’)] that
$P_{\lambda/\mu}(t^{k},\ldots,t^{n-1})=0$, showing (1). Approximating
$\mu\in\operatorname{\mathbb{GT}}_{\infty}$ with
$(\mu_{1},\ldots,\mu_{n})\in\operatorname{\mathbb{GT}}_{n}$ and invoking 4.5
yields (2). ∎

4.6 could also be shown by the explicit formula (4.7), but as the above proof
shows, it in fact requires only the very basic properties of symmetric
functions.

######  Proposition 4.7.

Every extreme coherent system is given by $(M^{\mu}_{n})_{n\geq 1}$ for some
$\mu\in\operatorname{\mathbb{GT}}_{\infty}$.

###### Proof.

Let $(M_{n})_{n\geq 1}$ be an extreme coherent system and $(\mu(n))_{n\geq 1}$
be a regular sequence converging to it, the existence of which is guaranteed
by 4.4. We wish to find $\mu\in\operatorname{\mathbb{GT}}_{\infty}$ such that

$\lim_{n\to\infty}L^{n}_{k}(\mu(n),\lambda)=M^{\mu}_{k}(\lambda)$ (4.14)

for all $k$ and $\lambda\in\operatorname{\mathbb{GT}}_{k}$, and will construct
$\mu$ as a limit of the signatures $\mu(n)$.

Our first step is to show the sequence of first parts $(\mu_{1}(n))_{n\geq 1}$
is bounded above (and hence all other $(\mu_{i}(n))_{n\geq 1}$ are as well).
Suppose for the sake of contradiction that this is not the case. Then there is
a subsequence $(\mu_{1}(n_{j}))_{j\geq 1}$ of $(\mu_{1}(n))_{n\geq 1}$ for
which $\mu_{1}(n_{j})\to\infty$. We claim that for any $k$ and
$\lambda\in\operatorname{\mathbb{GT}}_{k}$,

$\lim_{j\to\infty}L^{n_{j}}_{k}(\mu^{(}n_{j}),\lambda)=0.$ (4.15)

This suffices for the contradiction, as then (4.15) holds also with $n_{j}$
replaced by $n$ by regularity of $(\mu(n))_{n\geq 1}$, therefore the sequence
of probability measures $L^{n}_{k}(\mu(n),\cdot)$ converges to the zero
measure, which contradicts the definition of regular sequence. So let us prove
(4.15), and to declutter notation let us without loss of generality denote the
subsequence by $(\mu(n))_{n\geq 1}$ as well.

We claim there exists a constant $C_{k}$ such that for all $J\geq 1$ and
$\nu\in\operatorname{\mathbb{GT}}_{k+J}$,

$\left|t^{m_{x}(\nu)m_{x}(\lambda)}\;_{3}\bar{\phi}_{2}\left(\begin{matrix}t^{-m_{x}(\lambda)};t^{-m_{x}(\nu)},0\\\
t^{1+\nu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\nu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\end{matrix};t,t\right)\right|\leq
C_{k}$ (4.16)

For fixed $\lambda$, $1+\nu_{x+1}^{\prime}-\lambda_{x}^{\prime}$ and
$1+J-\nu_{x}^{\prime}+\lambda_{x+1}^{\prime}$ are both bounded below
independent of $\nu$ by $1-k$. This gives an upper bound on the factors
$(bt^{\ell};t)_{m_{x}(\lambda)-\ell},0\leq\ell\leq m_{x}(\lambda)$ where
$b\in\\{t^{1+\nu_{x+1}^{\prime}-\lambda_{x}^{\prime}},t^{1+J-\nu_{x}^{\prime}+\lambda_{x+1}^{\prime}}\\}$
which appear in the sum expansion (3.3) of (4.16). The term
$t^{m_{x}(\nu)m_{x}(\lambda)}(t^{-m_{x}(\nu)};t)_{\ell}$ is likewise bounded
above independent of $\nu$. Because $m_{x}(\lambda)$ and
$\lambda_{x}^{\prime}$ can only take finitely many values, the claim follows.
Furthermore, the LHS of (4.16) is simply $1$ whenever $m_{x}(\lambda)=0$,
which is true for all but finitely many $x$. Plugging this bound into 4.1
yields

$L^{n}_{k}(\mu(n),\lambda)\leq\frac{C_{k}^{k}}{\begin{bmatrix}n\\\
k\end{bmatrix}\prod_{i\in\mathbb{Z}}(t;t)_{m_{i}(\lambda)}}\prod_{x\in\mathbb{Z}}t^{(k-\lambda_{x}^{\prime})(\mu(n)_{x}^{\prime}-\lambda_{x}^{\prime})}.$
(4.17)

For $\lambda_{1}<x\leq\mu(n)_{1}$, one has
$t^{(k-\lambda_{x}^{\prime})(\mu(n)_{x}^{\prime}-\lambda_{x}^{\prime})}\leq
t^{k}<1$, and our claim (4.15) follows.

Now, suppose for the sake of contradiction that there exists $k$ for which
$(\mu(n)_{k})_{n\geq 1}$ is not bounded below. Then for any
$\lambda\in\operatorname{\mathbb{GT}}_{k}$, there are infinitely many $n$ for
which $\mu(n)_{k}<\lambda_{k}$ and consequently
$\mu(n)_{x}^{\prime}<\lambda_{x}^{\prime}=k$ for $x=\lambda_{k}$. By 4.6,
$L^{n}_{k}(\mu(n),\lambda)=0$ for all such $n$, therefore
$L^{n}_{k}(\mu(n),\lambda)\to 0$ as $n\to\infty$ since $(\mu(n))_{n\geq 1}$ is
a regular sequence. This is a contradiction, therefore $(\mu(n)_{k})_{n\geq
1}$ is bounded below for each $k$.

Since $(\mu(n)_{k})_{n\geq 1}$ is bounded above and below for each $k$, there
is a subsequence on which these converge, and by a diagonalization argument
there exists a subsequence $(\mu(n_{j}))_{j\geq 1}$ on which $\mu(n_{j})_{k}$
converges for every $k$. Letting $\mu_{i}=\lim_{j\to\infty}\mu(n_{j})_{i}$ and
$\mu=(\mu_{1},\mu_{2},\ldots)\in\operatorname{\mathbb{GT}}_{\infty}$, we have
by 4.5 that

$\lim_{j\to\infty}L_{k}^{n_{j}}(\mu(n_{j}),\lambda)=M^{\mu}_{k}(\lambda)$

for each $\lambda\in\operatorname{\mathbb{GT}}_{k}$. Since
$\lim_{n\to\infty}L_{k}^{n}(\mu(n),\lambda)$ exists by the definition of
regular sequence, it must also be equal to $M^{\mu}_{k}(\lambda)$. This shows
(4.14), completing the proof. ∎

For the other direction, 4.10, we will need the basic fact that general
coherent systems are convex combinations of extreme ones.

######  Proposition 4.8.

For any coherent system $(M_{n})_{n\geq 1}$ on $\mathscr{G}_{t}$, there exists
a Borel333The topology on $\partial\mathscr{G}_{t}$ here is the following. For
each $n$, the set of probability measures on $\operatorname{\mathbb{GT}}_{n}$
inherits a topology from the product topology on $\mathbb{R}^{\infty}$ by
viewing the measures as functions, which gives a topology on the inverse limit
$\partial\mathscr{G}_{t}$. measure $\pi$ on $\partial\mathscr{G}_{t}$ such
that

$M_{k}=\int_{M^{\prime}\in\partial\mathscr{G}_{t}}M_{k}^{\prime}\pi(dM^{\prime})$

for each $k$, where $M^{\prime}$ is shorthand for a coherent system
$(M^{\prime}_{n})_{n\geq 1}$.

###### Proof.

Follows from [33, Theorem 9.2]. ∎

It will also be necessary to put a topology on
$\operatorname{\mathbb{GT}}_{\infty}$, namely the one inherited from the
product topology on $\mathbb{Z}^{\infty}$ where $\mathbb{Z}$ is equipped with
the cofinite topology. The following lemma shows that these natural choices of
topology on $\operatorname{\mathbb{GT}}_{\infty}$ and
$\partial\mathscr{G}_{t}$ are compatible.

######  Lemma 4.9.

The map

$\displaystyle f:\operatorname{\mathbb{GT}}_{\infty}$
$\displaystyle\to\mathcal{M}(\partial\mathscr{G}_{t})$ $\displaystyle\mu$
$\displaystyle\mapsto(M^{\mu}_{n})_{n\geq 1}$

is continuous, hence in particular Borel.

###### Proof.

Since $\operatorname{\mathbb{GT}}_{\infty}$ is first-countable, to show $f$ is
continuous it suffices to show it preserves limits of sequences. Hence we must
show that for any $\mu\in\operatorname{\mathbb{GT}}_{\infty}$, if
$\nu^{(1)},\nu^{(2)},\ldots\in\operatorname{\mathbb{GT}}_{\infty}$ and
$\nu^{(k)}_{i}\to\mu_{i}$ for all $i$, then $M^{\nu^{(k)}}_{n}\to M^{\mu}_{n}$
pointwise as functions on $\operatorname{\mathbb{GT}}_{n}$. This follows
straightforwardly from the explicit formula (4.7) of 4.2. ∎

######  Proposition 4.10.

For every $\mu\in\operatorname{\mathbb{GT}}_{\infty}$, the coherent system
$(M^{\mu}_{n})_{n\geq 1}$ is extreme.

###### Proof.

Fix $\mu\in\operatorname{\mathbb{GT}}_{\infty}$. By 4.7,there is a Borel
measure $\pi\in\mathcal{M}(\partial\mathscr{G}_{t})$.

$M^{\mu}_{k}=\int_{M^{\prime}\in\partial\mathscr{G}_{t}}M_{k}^{\prime}\pi(dM^{\prime})=\int_{\nu\in\operatorname{\mathbb{GT}}_{\infty}}M^{\nu}_{k}(\iota_{*}\pi)(d\nu)$
(4.18)

where
$\iota:\partial\mathscr{G}_{t}\hookrightarrow\operatorname{\mathbb{GT}}_{\infty}$
is the inclusion guaranteed by 4.7. Because $f\circ\iota=\operatorname{Id}$
and $f$ is Borel, $\iota$ is a Borel isomorphism onto its image, hence
$\iota_{*}\pi$ is a Borel measure in the topology on
$\operatorname{\mathbb{GT}}_{\infty}$ above.

We first claim that $\iota_{*}\pi$ is supported on

$S_{\leq\mu}:=\\{\nu\in\operatorname{\mathbb{GT}}_{\infty}:\nu_{i}\leq\mu_{i}\text{
for all }i\\}.$

Suppose not. Since

$\operatorname{\mathbb{GT}}_{\infty}\setminus S_{\leq\mu}=\bigcup_{k\geq
1}\\{\nu\in\operatorname{\mathbb{GT}}_{\infty}:\nu_{i}>\mu_{i}\text{ for at
least one }1\leq i\leq k\\}$

and

$\\{\nu\in\operatorname{\mathbb{GT}}_{\infty}:\nu_{i}>\mu_{i}\text{ for at
least one }1\leq i\leq
k\\}=\bigcup_{\begin{subarray}{c}\lambda\in\operatorname{\mathbb{GT}}_{k}:\\\
\exists i\text{ s.t.
}\lambda_{i}>\mu_{i}\end{subarray}}\\{\nu\in\operatorname{\mathbb{GT}}_{\infty}:\nu_{i}=\lambda_{i}\text{
for all }1\leq i\leq k\\},$

if $(\iota_{*}\pi)(\operatorname{\mathbb{GT}}_{\infty}\setminus
S_{\leq\mu})>0$ then there exists $k$ and
$\lambda\in\operatorname{\mathbb{GT}}_{k}$ such that

$(\iota_{*}\pi)(\\{\nu\in\operatorname{\mathbb{GT}}_{\infty}:\nu_{i}=\lambda_{i}\text{
for all }1\leq i\leq k\\})>0.$ (4.19)

Denoting the set in (4.19) by
$S_{k}(\lambda)\subset\operatorname{\mathbb{GT}}_{\infty}$, we have

$M_{k}^{\mu}(\lambda_{1},\ldots,\lambda_{k})=\int_{\nu\in
S_{k}(\lambda)}M_{k}^{\nu}(\lambda_{1},\ldots,\lambda_{k})(\iota_{*}\pi)(d\nu)+\int_{\nu\in\operatorname{\mathbb{GT}}_{\infty}\setminus
S_{k}(\lambda)}M_{k}^{\nu}(\lambda_{1},\ldots,\lambda_{k})(\iota_{*}\pi)(d\nu).$
(4.20)

The LHS is $0$ by 4.6. If $\nu\in S_{k}(\lambda)$, then the only factor in

$M_{k}^{\nu}(\lambda_{1},\ldots,\lambda_{k})=\begin{bmatrix}k\\\
\lambda\end{bmatrix}_{t}\prod_{x\in\mathbb{Z}_{\geq\lambda_{k}}}t^{(\nu^{\prime}_{x}-\lambda^{\prime}_{x})(k-\lambda^{\prime}_{x})}(t^{1+\nu^{\prime}_{x}-\lambda^{\prime}_{x}};t)_{m_{x}(\lambda)}$

which depends on $\nu$ is
$(t^{1+\nu^{\prime}_{\lambda_{k}}-k};t)_{m_{\lambda_{k}}(\lambda)}$, which is
clearly bounded below by $(t;t)_{\infty}$. Hence the RHS of (4.20) is bounded
below by

$(\iota_{*}\pi)(S_{k}(\lambda))(t;t)_{\infty}\begin{bmatrix}k\\\
\lambda\end{bmatrix}_{t}>0,$

a contradiction. Therefore $\iota_{*}\pi$ is indeed supported on
$S_{\leq\mu}$.

For each $k\geq 1$ we may decompose

$S_{\leq\mu}=\left(S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k})\right)\sqcup\left(S_{\leq\mu}\cap\left(S_{k}(\mu_{1},\ldots,\mu_{k})\right)^{c}\right)$

into those signatures which agree with $\mu$ on the first $k$ coordinates and
those which do not, and

$M^{\mu}_{k}(\mu_{1},\ldots,\mu_{k})=\int_{\nu\in S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k})}M^{\nu}_{k}(\mu_{1},\ldots,\mu_{k})(\iota_{*}\pi)(d\nu)\\\
+\int_{\nu\in
S_{\leq\mu}\cap(S_{k}(\mu_{1},\ldots,\mu_{k})^{c})}M^{\nu}_{k}(\mu_{1},\ldots,\mu_{k})(\iota_{*}\pi)(d\nu).$
(4.21)

The second integral in (4.21) is always $0$ by 4.6. If $\nu\in S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k})$ then $\nu_{x}^{\prime}=\mu_{x}^{\prime}$ for
$x>\mu_{k}$ and $\nu_{x}^{\prime}\leq\mu_{x}^{\prime}$ when $x=\mu_{k}$. Hence

$(t^{1+\nu_{x}^{\prime}-k};t)_{m_{x}(\mu_{1},\ldots,\mu_{k})}\leq(t^{1+\mu_{x}^{\prime}-k};t)_{m_{x}(\mu_{1},\ldots,\mu_{k})}$

for all $x$, and all other factors in (4.7) are the same for
$M_{k}^{\nu}(\mu_{1},\ldots,\mu_{k})$ and
$M_{k}^{\mu}(\mu_{1},\ldots,\mu_{k})$, therefore

$M_{k}^{\nu}(\mu_{1},\ldots,\mu_{k})\leq
M_{k}^{\mu}(\mu_{1},\ldots,\mu_{k})\text{ for all }\nu\in S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k}).$

Hence (4.21) reduces to

$M^{\mu}_{k}(\mu_{1},\ldots,\mu_{k})\leq
M^{\mu}_{k}(\mu_{1},\ldots,\mu_{k})\cdot(\iota_{*}\pi)(S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k})).$ (4.22)

Since $M^{\mu}_{k}(\mu_{1},\ldots,\mu_{k})>0$ by (4.7), it follows that

$(\iota_{*}\pi)(S_{\leq\mu}\cap S_{k}(\mu_{1},\ldots,\mu_{k}))=1.$

Since this is true for all $k$ and $\bigcap_{k}\left(S_{\leq\mu}\cap
S_{k}(\mu_{1},\ldots,\mu_{k})\right)=\\{\mu\\}$, it follows that
$(\iota_{*}\pi)(\\{\mu\\})=1$, i.e. $\iota_{*}\pi$ is the delta mass at $\mu$.
Hence $(M^{\mu}_{n})_{n\geq 1}$ is an extreme coherent system, completing the
proof. ∎

##  5 Infinite $p$-adic random matrices and corners

In this section, we turn to $p$-adic random matrix theory and prove Section
1.3 and Section 1.4. We will first give the basic setup of $p$-adic random
matrices and the key result 5.2 which relates the operations of removing rows
and columns to Hall-Littlewood polynomials. In Section 5.2 we prove auxiliary
boundary results on a slightly more complicated branching graph which extends
the one in the previous section, which are tailored to the random matrix
corner situation. We then use these to deduce the result Section 1.3, that
extreme bi-invariant measures on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ are parametrized by
the set $\operatorname{\overline{\mathbb{GT}}}_{\infty}$ defined in 14 below,
from the parametrization of the boundary of this augmented branching graph by
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ (5.4).

###  5.1 $p$-adic background.

The following basic background is more or less quoted from [39] and is a
condensed version of the exposition in [17, §2], to which we refer any reader
desiring a more detailed introduction to $p$-adic numbers. Fix a prime $p$.
Any nonzero rational number $r\in\mathbb{Q}^{\times}$ may be written as
$r=p^{k}(a/b)$ with $k\in\mathbb{Z}$ and $a,b$ coprime to $p$. Define
$|\cdot|:\mathbb{Q}\to\mathbb{R}$ by setting $|r|_{p}=p^{-k}$ for $r$ as
before, and $|0|_{p}=0$. Then $|\cdot|_{p}$ defines a norm on $\mathbb{Q}$ and
$d_{p}(x,y):=|x-y|_{p}$ defines a metric. We define the _field of $p$-adic
numbers_ $\mathbb{Q}_{p}$ to be the completion of $\mathbb{Q}$ with respect to
this metric, and the _$p$ -adic integers_ $\mathbb{Z}_{p}$ to be the unit ball
$\\{x\in\mathbb{Q}_{p}:|x|_{p}\leq 1\\}$. It is not hard to check that
$\mathbb{Z}_{p}$ is a subring of $\mathbb{Q}_{p}$. We remark that
$\mathbb{Z}_{p}$ may be alternatively defined as the inverse limit of the
system
$\ldots\to\mathbb{Z}/p^{n+1}\mathbb{Z}\to\mathbb{Z}/p^{n}\mathbb{Z}\to\cdots\to\mathbb{Z}/p\mathbb{Z}\to
0$, and that $\mathbb{Z}$ naturally includes into $\mathbb{Z}_{p}$.

$\mathbb{Q}_{p}$ is noncompact but is equipped with a left- and right-
invariant (additive) Haar measure; this measure is unique if we normalize so
that the compact subgroup $\mathbb{Z}_{p}$ has measure $1$, and we denote it
by $\mu_{Haar}$. The restriction of this measure to $\mathbb{Z}_{p}$ is the
unique Haar probability measure on $\mathbb{Z}_{p}$, and is explicitly
characterized by the fact that its pushforward under any map
$r_{n}:\mathbb{Z}_{p}\to\mathbb{Z}/p^{n}\mathbb{Z}$ is the uniform probability
measure. For concreteness, it is often useful to view elements of
$\mathbb{Z}_{p}$ as ‘power series in $p$’ $a_{0}+a_{1}p+a_{2}p^{2}+\ldots$,
with $a_{i}\in\\{0,\ldots,p-1\\}$; clearly these specify a coherent sequence
of elements of $\mathbb{Z}/p^{n}\mathbb{Z}$ for each $n$. The Haar probability
measure then has the alternate explicit description that each $a_{i}$ is iid
uniformly random from $\\{0,\ldots,p-1\\}$. Additionally, $\mathbb{Q}_{p}$ is
isomorphic to the ring of Laurent series in $p$, defined in exactly the same
way.

$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$
acts on $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ by left- and right
multiplication. The orbits of this action are parametrized by signatures with
possibly infinite parts, which we now define formally.

######  Definition 14.

For $n\in\mathbb{Z}_{\geq 1}$, we let

$\operatorname{\overline{\mathbb{GT}}}_{n}:=\\{(\lambda_{1},\ldots,\lambda_{n})\in(\mathbb{Z}\cup\\{-\infty\\})^{n}:\lambda_{1}\geq\ldots\geq\lambda_{n}\\},$

where we take $-\infty<a$ for all $a\in\mathbb{Z}$, and refer to elements of
$\operatorname{\overline{\mathbb{GT}}}_{n}$ as _extended signatures_. The
definition of $\operatorname{\overline{\mathbb{GT}}}_{\infty}$ is exactly
analogous. For $0\leq k\leq n$, we denote by
$\operatorname{\overline{\mathbb{GT}}}_{n}^{(k)}\subset\operatorname{\overline{\mathbb{GT}}}_{n}$
the set of all extended signatures with exactly $k$ integer parts and the rest
equal to $-\infty$. For
$\lambda\in\operatorname{\overline{\mathbb{GT}}}_{n}^{(k)}$, we denote by
$\lambda^{*}\in\operatorname{\mathbb{GT}}_{k}$ the signature given by its
integer parts.

The parametrization, stated below, is often called Smith normal form.

######  Proposition 5.1.

Let $n\leq m$. For any $A\in\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$,
there exists a unique $\lambda\in\operatorname{\overline{\mathbb{GT}}}_{n}$
for which there exist
$U\in\operatorname{GL}_{n}(\mathbb{Z}_{p}),V\in\operatorname{GL}_{m}(\mathbb{Z}_{p})$
such that $UAV=\operatorname{diag}_{n\times
m}(p^{-\lambda_{1}},\ldots,p^{-\lambda_{n}})$, where we formally take
$p^{\infty}=0$.

For any $\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\min(m,n)}$, it is
useful to consider the probability measure on $\operatorname{Mat}_{n\times
m}(\mathbb{Q}_{p})$ with distribution $U\operatorname{diag}_{n\times
m}(p^{-\lambda_{1}},\ldots,p^{-\lambda_{n}})V$ where
$U\in\operatorname{GL}_{n}(\mathbb{Z}_{p}),V\in\operatorname{GL}_{m}(\mathbb{Z}_{p})$
are Haar-distributed. By uniqueness of the Haar measure, this measure is the
unique one which is left- and right-invariant under multiplication by
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$ and
$\operatorname{GL}_{m}(\mathbb{Z}_{p})$ respectively.

######  Definition 15.

We denote the extended signature in 5.1 by
$\operatorname{SN}(A)\in\operatorname{\overline{\mathbb{GT}}}_{\min(m,n)}$,
and denote this signature padded with infinitely many parts equal to $-\infty$
by $\operatorname{ESN}(A)\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$.
We refer to the finite parts of either signature as the _singular numbers_ of
$A$.

The reason for padding with $-\infty$ is to allow us to treat matrices of
different sizes on equal footing, essentially viewing them as corners of a
large matrix of low rank. It is somewhat unwieldy but seemed to be the least
awkward formalism for the problem at hand.

######  Remark 8.

We have defined singular numbers with the opposite sign convention as [39]
(though the same sign convention as [1, 12]) to match with the branching graph
notation of Section 4 and the latter references.

######  Proposition 5.2.

Let $n,m\geq 1$ be integers,
$\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ with
$\operatorname{len}(\mu^{*})\leq\min(m+1,n)$, let
$A\in\operatorname{Mat}_{n\times(m+1)}(\mathbb{Q}_{p})$ be distributed by the
unique bi-invariant measure with singular numbers $\mu$, and let $t=1/p$. If
$A^{\prime}\in\operatorname{Mat}_{n\times m}$ is the first $m$ columns of $A$,
then $\operatorname{ESN}(A^{\prime})$ is a random element of
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ with

$\Pr(\operatorname{ESN}(A^{\prime})=\lambda)=\begin{cases}\frac{Q_{-\lambda^{*}/-\mu^{*}}(t^{m+1-k})P_{-\lambda^{*}}(1,\ldots,t^{k-1})}{P_{-\mu^{*}}(1,\ldots,t^{k-1})\Pi(t^{m+1-k};1,\ldots,t^{k-1})}&\mu,\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(k)}\text{
for some }0\leq k\leq\min(m,n)\\\
P_{\mu^{*}/\lambda^{*}}(t^{m})\frac{P_{\lambda^{*}}(1,\ldots,t^{m-1})}{P_{\mu^{*}}(1,\ldots,t^{m})}&\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(m+1)},\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(m)}\\\
0&\text{otherwise}\end{cases}$ (5.1)

for any $\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$.

###### Proof.

In the case where $\operatorname{len}(\mu^{*})=\min(m+1,n)$ so that $A$ is
full-rank, the result follows by applying [39, Theorem 1.3, Part 2] (taking
care that the singular numbers in that paper are the negatives of the singular
numbers here). The non full-rank case
$\operatorname{len}(\mu^{*})<\min(m+1,n)$ follows from the full-rank case with
$m+1>n$, as in this case the rank of $A$ does not change after removing the
$(m+1)^{th}$ column. ∎

Because $\operatorname{ESN}(A)=\operatorname{ESN}(A^{T})$, 5.2 obviously holds
for removing rows rather than columns after appropriately relabeling the
indices. By relating matrix corners to Hall-Littlewood polynomials, 5.2
provides the key to applying the results on Hall-Littlewood branching graphs
to study $p$-adic random matrices. In the second case of the transition
probabilities in (5.1), one immediately recognizes the cotransition
probabilities of Section 4. However, one now has two added features not
present in that section: (1) the signatures may have infinite parts, and (2)
with matrices one may remove either rows or columns, so there are in fact two
(commuting) corner maps. In the next subsection, we augment the branching
graph formalism and results of Section 4 to handle this more complicated
setup. However, let us first introduce the setup of infinite matrices.

######  Definition 16.

$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$ is the direct limit
$\varinjlim\operatorname{GL}_{N}(\mathbb{Z}_{p})$ with respect to inclusions

$\displaystyle\operatorname{GL}_{N}(\mathbb{Z}_{p})$
$\displaystyle\hookrightarrow\operatorname{GL}_{N+1}(\mathbb{Z}_{p})$
$\displaystyle A$ $\displaystyle\mapsto\begin{pmatrix}A&0\\\ 0&1\end{pmatrix}$

Equivalently, $\operatorname{GL}_{\infty}(\mathbb{Z}_{p})=\bigcup_{N\geq
1}\operatorname{GL}_{N}(\mathbb{Z}_{p})$ where we identify
$\operatorname{GL}_{N}(\mathbb{Z}_{p})$ with the group of infinite matrices
for which the top left $N\times N$ corner is an element of
$\operatorname{GL}_{N}(\mathbb{Z}_{p})$ and all other entries are $1$ on the
diagonal and $0$ off the diagonal.

The definition

$\operatorname{Mat}_{n\times
m}(\mathbb{Q}_{p}):=\left\\{Z=(Z_{ij})_{\begin{subarray}{c}1\leq i\leq n\\\
1\leq j\leq m\end{subarray}}:Z_{ij}\in\mathbb{Q}_{p}\right\\}.$

still makes sense when $n$ or $m$ is equal to $\infty$ by replacing $1\leq
i\leq n$ with $i\in\mathbb{Z}_{\geq 1}$ and similarly for $m$. When $n$ or $m$
is $\infty$, $\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$ clearly acts on this
space on the appropriate side.

###  5.2 Auxiliary boundary results and proof of Section 1.3.

In this subsection we prove a similar result to Section 1.2, 5.3, and deduce
an extension to a ‘two-dimensional’ version of the branching graph
$\mathscr{G}_{t}$ in 5.4.

######  Definition 17.

For each $k\geq 1$, we define a graded graph

$\mathscr{G}_{t}^{(k)}=\bigsqcup_{n\geq 1}\mathscr{G}_{t}^{(k)}(n)$

with vertex set at each level given by
$\mathscr{G}_{t}^{(k)}(n)=\operatorname{\mathbb{GT}}_{k}$. Edges are only
between adjacent levels, and to each edge from
$\nu\in\mathscr{G}_{t}^{(k)}(n+1)$ to $\lambda\in\mathscr{G}_{t}^{(k)}(n)$ is
associated a cotransition probability

$\tilde{L}^{n+1}_{n}(\nu,\lambda)=Q_{-\lambda/-\nu}(t^{n})\frac{P_{-\lambda}(1,t,\ldots,t^{k-1})}{P_{-\nu}(1,t,\ldots,t^{k-1})\Pi(1,t,\ldots,t^{k-1};t^{n})}.$

We define $\tilde{L}^{m}_{n}=\tilde{L}^{n+1}_{n}\cdots\tilde{L}^{m}_{m-1}$ for
general $1\leq n<m<\infty$ as before.

The next result is a version of Section 1.2 for this smaller branching graph
$\mathscr{G}_{t}^{(k)}$. Recall the definition of boundary from earlier in
this section.

######  Theorem 5.3.

For any $t\in(0,1)$, the boundary $\partial\mathscr{G}_{t}^{(k)}$ is naturally
in bijection with $\operatorname{\mathbb{GT}}_{k}$. Under this bijection,
$\mu\in\operatorname{\mathbb{GT}}_{k}$ corresponds to the coherent system
$(M_{n}^{\mu})_{n\geq 1}$ defined explicitly by

$M_{n}^{\mu}(\lambda)=Q_{-\lambda/-\mu}(t^{n},t^{n+1},\ldots)\frac{P_{-\lambda}(1,t,\ldots,t^{k-1})}{P_{-\mu}(1,t,\ldots,t^{k-1})\Pi(1,t,\ldots,t^{k-1};t^{n},t^{n+1},\ldots)}$
(5.2)

for $\lambda\in\operatorname{\mathbb{GT}}_{k}$.

Note we have simultaneously suppressed the $k$-dependence in our notation for
the measure $M_{n}^{\mu}$ on $\operatorname{\mathbb{GT}}_{k}$ and abused
notation by using the same for measures on $\mathscr{G}_{t}$ and
$\mathscr{G}_{t}^{(k)}$, but there is no ambiguity if one knows the length of
$\mu$. The proof of 5.3 is an easier version of the proof of Section 1.2, so
we simply give a sketch and outline the differences.

###### Proof.

We first prove that every extreme coherent system is of the form (5.2) for
some $\mu\in\operatorname{\mathbb{GT}}_{k}$. The analogue of 4.4 similarly
follows from the general result [32, Theorem 6.1], so there exists a regular
sequence $(\mu(n))_{n\geq 1}$ approximating any extreme coherent system. Using
the explicit formula (3.7) of 3.2, a naive bound as in the proof of 4.7
establishes that $\mu(n)_{1}$ is bounded above.

The analogue of 4.6, namely that $\tilde{L}^{m}_{n}(\mu,\lambda)=0$ and
$M_{n}^{\mu}(\lambda)=0$ if there exists an $x$ for which
$\lambda_{x}^{\prime}>\mu_{x}^{\prime}$, holds similarly by the branching
rule. Using this one obtains that a regular sequence $(\mu(n))_{n\geq 1}$ must
have last parts $\mu(n)_{k}$ bounded below. Together with the upper bound this
yields that $(\mu(n))_{n\geq 1}$ has a convergent subsequence, where here
convergence simply means that all terms in the subsequence are equal to the
same $\mu\in\operatorname{\mathbb{GT}}_{k}$. It now follows as in the proof of
4.7 that in fact the coherent system approximated by $(\mu(n))_{n\geq 1}$ must
be $(M^{\mu}_{n})_{n\geq 1}$ for this $\mu$.

It remains to prove that every coherent system of the form (5.2) is in fact
extreme. The proof is the same as that of 4.10 using the above analogue of
4.6, except that no measure-theoretic details are necessary because the
decomposition of an arbitrary coherent system into extreme ones takes the form
of a sum over the countable set $\operatorname{\mathbb{GT}}_{k}$. ∎

For applications in the next section it is desirable to in some sense combine
$\mathscr{G}_{t}$ and $\mathscr{G}_{t}^{(k)}$ by working with extended
signatures. We wish to define a doubly-graded graph with cotransition
probabilities which generalize the earlier $L^{n+1}_{n},\tilde{L}^{n+1}_{n}$
and which correspond to the situation of removing rows and columns from a
matrix in 5.2.

######  Definition 18.

Define

$\widetilde{\mathscr{G}_{t}}=\bigsqcup_{m,n\geq
1}\widetilde{\mathscr{G}_{t}}(m,n)$

with
$\widetilde{\mathscr{G}_{t}}(m,n)=\operatorname{\overline{\mathbb{GT}}}_{\infty}$
for each $m,n$, and edges from $\widetilde{\mathscr{G}_{t}}(m+1,n)$ to
$\widetilde{\mathscr{G}_{t}}(m,n)$ with weights

$L^{m+1,n}_{m,n}(\mu,\lambda)=\begin{cases}\frac{Q_{-\lambda^{*}/-\mu^{*}}(t^{m+1-k})P_{-\lambda^{*}}(1,\ldots,t^{k-1})}{P_{-\mu^{*}}(1,\ldots,t^{k-1})\Pi(t^{m+1-k};1,\ldots,t^{k-1})}&\mu,\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(k)}\text{
for some }0\leq k\leq\min(m,n)\\\
P_{\mu^{*}/\lambda^{*}}(t^{m})\frac{P_{\lambda^{*}}(1,\ldots,t^{m-1})}{P_{\mu^{*}}(1,\ldots,t^{m})}&\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(m+1)},\lambda\in\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(m)}\\\
0&\text{otherwise}\end{cases}$ (5.3)

and edges from $\widetilde{\mathscr{G}_{t}}(m,n+1)$ to
$\widetilde{\mathscr{G}_{t}}(m,n)$ with weights
$L^{m,n+1}_{m,n}(\mu,\lambda)=L^{n+1,m}_{n,m}(\mu,\lambda)$.

It follows immediately from the Cauchy identity 2.2 that

$L^{m+1,n}_{m,n}L^{m+1,n+1}_{m+1,n}=L^{m,n+1}_{m,n}L^{m+1,n+1}_{m,n+1},$

so there is no ambiguity in defining coherent systems of probability measures
on $\widetilde{\mathscr{G}_{t}}$.

######  Theorem 5.4.

For $t\in(0,1)$, the boundary $\partial\widetilde{\mathscr{G}_{t}}$ is in
bijection with $\operatorname{\overline{\mathbb{GT}}}_{\infty}$. The extreme
coherent system $(M^{\mu}_{m,n})_{m,n\geq 1}$ corresponding to
$\mu\in\operatorname{\mathbb{GT}}_{\infty}$ is determined by

$M^{\mu}_{m,n}(\nu)=\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}}M^{\mu}_{n}(\lambda)M^{\lambda}_{m-n+1}(\nu^{*})$

for $m\geq n$ and hence for all $m,n$ by coherency. The extreme coherent
system corresponding to $\mu\in\operatorname{\mathbb{GT}}_{\infty}^{(k)}$ is
determined by

$M^{\mu}_{m,n}(\nu)=\sum_{\lambda\in\operatorname{\mathbb{GT}}_{k}}M^{\mu^{*}}_{n-k+1}(\lambda)M^{\lambda}_{m-k+1}(\nu^{*})$

for $m,n\geq k$ and hence for all $m,n$ by coherency.

###### Proof.

First note that every coherent system on $\widetilde{\mathscr{G}_{t}}$ is
determined by a sequence of coherent systems on the subgraphs with vertex sets

$\bigsqcup_{m\geq n}\widetilde{\mathscr{G}_{t}}(m,n)$ (5.4)

for $n\geq 1$, which are themselves coherent with one another under the links
$L^{m,n+1}_{m,n}$. By the definition of the cotransition probabilities (5.3),
a coherent system on (5.4) must decompose as a convex combination of $n+1$
coherent systems, each one having all measures supported on
$\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(k)}$ for $0\leq k\leq n$.
Hence extreme coherent systems on (5.4) are parametrized by
$\operatorname{\overline{\mathbb{GT}}}_{k}$ by applying 5.3 for each $k$.

It follows by the above-mentioned commutativity
$L^{m+1,n}_{m,n}L^{m+1,n+1}_{m+1,n}=L^{m,n+1}_{m,n}L^{m+1,n+1}_{m,n+1}$ that
given a coherent system $(M_{m})_{m\geq n}$ on the graph (5.4),
$(M_{m}L^{m,n}_{m,n-1})_{m\geq n}$ is a coherent system on

$\bigsqcup_{m\geq n}\widetilde{\mathscr{G}_{t}}(m,n-1).$

Since $L^{m,n}_{m,n-1}$ takes coherent systems to coherent systems, by
decomposing these into extreme coherent systems it induces a map
$\mathcal{M}(\operatorname{\overline{\mathbb{GT}}}_{n})\to\mathcal{M}(\operatorname{\overline{\mathbb{GT}}}_{n-1})$
between spaces of probability measures on the respective boundaries, i.e. a
Markov kernel. It follows from the explicit formulas (5.2), (5.3) and the
Cauchy identity 2.2 that this Markov map is itself given by $L^{m,n}_{m,n-1}$
on the appropriately restricted domain, after identifying
$\operatorname{\overline{\mathbb{GT}}}_{n}$ and
$\operatorname{\overline{\mathbb{GT}}}_{n-1}$ as subsets of
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ in the obvious way.

Hence $\partial\widetilde{\mathscr{G}_{t}}$ is in bijection with coherent
systems on the graph with vertex set

$\bigsqcup_{n\geq 1}\operatorname{\overline{\mathbb{GT}}}_{n}$

and edges between $n^{th}$ and $(n-1)^{\text{st}}$ level given by
$L^{m,n}_{m,n-1}$ for any $m\geq n$ (note the these links are independent of
$m\geq n$ by (5.3)). The boundary of this graph is classified by
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ by combining Section 1.2 (for
coherent systems supported on $\operatorname{\mathbb{GT}}_{\infty}$) and 5.3
(for coherent systems supported on
$\operatorname{\overline{\mathbb{GT}}}_{\infty}^{(k)}$), and the explicit
coherent systems in the statement follow from the above computations. ∎

###### Proof of Section 1.3.

Any
$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})\times\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$-invariant
measure on $\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ is
uniquely determined by its marginals on $m\times n$ truncations for finite
$m,n$, which are each
$\operatorname{GL}_{m}(\mathbb{Z}_{p})\times\operatorname{GL}_{n}(\mathbb{Z}_{p})$-invariant.
The
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$-invariant
probability measures on $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ are
in bijection with probability measures on
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ supported on signatures with
at most $\min(m,n)$ finite parts. Hence removing a row (resp. column) induces
a Markov kernel
$\mathcal{M}(\operatorname{\overline{\mathbb{GT}}}_{\infty})\to\mathcal{M}(\operatorname{\overline{\mathbb{GT}}}_{\infty})$,
and by 5.2 this Markov kernel is exactly $L^{m,n}_{m,n-1}$ (resp.
$L^{m,n}_{m-1,n}$). Hence 5.4 yields that the set of extreme
$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})\times\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$-invariant
measures on $\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ is in
bijection with $\operatorname{\overline{\mathbb{GT}}}_{\infty}$. Here the
measure $E_{\mu}$ corresponding to $\mu$ is determined by the fact that each
$m\times n$ corner has singular numbers distributed by the measure
$M_{m,n}^{\mu}$ in 5.4. ∎

We have shown that the extreme bi-invariant measures are parametrized somehow
by $\operatorname{\overline{\mathbb{GT}}}_{\infty}$, but in [12] the measure
corresponding to a given
$\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ is defined quite
differently, and it is not at all clear a priori that it is the same as our
measure $E_{\mu}$. Let us describe these measures.

In the finite or infinite setting, there are two natural families of random
matrices in $\operatorname{Mat}_{n\times m}(\mathbb{Q}_{p})$ which are
invariant under the natural action of
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$:

  * •

(Haar) $p^{-k}Z$, where $k\in\mathbb{Z}\cup\\{-\infty\\}$ and $Z$ has iid
entries distributed by the additive Haar measure on $\mathbb{Z}_{p}$.

  * •

(Nonsymmetric Wishart-type) $p^{-k}X^{T}Y$, where
$X\in\mathbb{Z}_{p}^{n},Y\in\mathbb{Z}_{p}^{m}$ have iid additive Haar
entries.

One can of course obtain invariant measures by summing the above random
matrices, which motivates the following class of measures.

######  Definition 19.

Let $\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$, and let
$\mu_{\infty}:=\lim_{\ell\to\infty}\mu_{\ell}\in\mathbb{Z}\cup\\{-\infty\\}$.
Let $X^{(\ell)}_{i},Y^{(\ell)}_{j},Z_{ij}$ be iid and distributed by the
additive Haar measure on $\mathbb{Z}_{p}$ for $i,j,\ell\geq 1$. Then we define
the measure $\tilde{E}_{\mu}$ on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ as the distribution
of the random matrix

$\left(\sum_{\ell:\mu_{\ell}>\mu_{\infty}}p^{-\mu_{\ell}}X^{(\ell)}_{i}Y^{(\ell)}_{j}+p^{-\mu_{\infty}}Z_{ij}\right)_{i,j\geq
1}.$

It is shown in [12, Theorem 1.3] that the
$\tilde{E}_{\mu},\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ are
exactly the extreme
$\operatorname{GL}_{\infty}(\mathbb{Z}_{p})\times\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$-invariant
measures on $\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$.

######  Proposition 5.5.

For any $\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$,
$\tilde{E}_{\mu}=E_{\mu}$.

###### Proof.

By combining Section 1.3 with the result [12, Theorem 1.3] that the
$\tilde{E}_{\mu}$ are exactly the extreme measures, we have that
$\\{\tilde{E}_{\mu}:\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}\\}=\\{E_{\mu}:\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}\\}$.
Hence for each $\mu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ there
exists $\nu\in\operatorname{\overline{\mathbb{GT}}}_{\infty}$ such that
$\tilde{E}_{\mu}=E_{\nu}$. Suppose for the sake of contradiction that
$\nu\neq\mu$. Let $k\geq 1$ be the smallest index for which
$\mu_{k}\neq\nu_{k}$, let

$f:\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})\to\operatorname{\overline{\mathbb{GT}}}_{k}$

be the map to the first $k$ singular numbers of the top left $k\times k$
corner, and let

$S^{(k)}_{\leq\mu}:=\\{\lambda\in\operatorname{\overline{\mathbb{GT}}}_{k}:\lambda_{i}\leq\mu_{i}\text{
for }1\leq i\leq k\\}.$

We claim that

$f_{*}(\tilde{E}_{\mu})\text{ is supported on }S^{(k)}_{\leq\mu}\text{ and
}(f_{*}(\tilde{E}_{\mu}))(\mu_{1},\ldots,\mu_{k})>0,$ (5.5)

and

$f_{*}(E_{\nu})\text{ is supported on }S^{(k)}_{\leq\nu}\text{ and
}(f_{*}(E_{\nu}))(\nu_{1},\ldots,\nu_{k})>0.$ (5.6)

The first, (5.5), follows straightforwardly from 19, while (5.6) follows from
5.4 and 4.6.

If $\mu_{k}>\nu_{k}$, then
$\operatorname{Supp}(f_{*}(\tilde{E}_{\mu}))\supsetneq\operatorname{Supp}(f_{*}(E_{\nu}))$,
while if $\mu_{k}<\nu_{k}$ then
$\operatorname{Supp}(f_{*}(\tilde{E}_{\mu}))\subsetneq\operatorname{Supp}(f_{*}(E_{\nu}))$,
contradicting the claim $f_{*}(\tilde{E}_{\mu})=f_{*}(E_{\nu})$. Therefore
there does not exist $k$ as above, so $\mu=\nu$, completing the proof. ∎

Combining 5.5 with Section 1.3 in fact provides a (quite indirect!)
computation of the singular numbers of $m\times n$ truncations of the infinite
matrices in 19.

######  Corollary 5.6.

The singular numbers of an $m\times n$ corner of an infinite matrix with
distribution $\tilde{E}_{\mu}$ are distributed by the measure $M_{m,n}^{\mu}$
of 5.4.

It seems possible that the summation which defines the measures
$M_{m,n}^{\mu}$ may be simplified to get more explicit formulas for the above
distributions, though we do not address this question here.

######  Remark 9.

There are several comments on the relation between our setup and that of [12]
which are worth highlighting:

  * •

We work over $\mathbb{Q}_{p}$ while [12] works over an arbitrary non-
Archimedean local field $F$. Such a field has a ring of integers
$\mathcal{O}_{F}$ playing the role of $\mathbb{Z}_{p}$ and a uniformizer
$\omega$ playing the role of $p$, and a finite residue field
$\mathcal{O}_{F}/\omega\mathcal{O}_{F}\cong\mathbb{F}_{q}$. Our results
transfer mutatis mutandis to this setting with $t=1/q$, as the only needed
input 5.2 transfers in view of [39, Remark 4].

  * •

While we simply prove a bijection, a short additonal argument shows that the
space of extreme invariant measures on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ is homeomorphic to
$\operatorname{\overline{\mathbb{GT}}}_{\infty}$ with natural topologies on
both spaces, see the proof of Theorem 1.3 of [12] for details.

  * •

We have used the language of extreme and ergodic measures interchangeably, but
for an explanation of how the extreme measures are exactly the ergodic ones in
the conventional sense, for this problem and more general versions, see [12,
Section 2.1].

######  Remark 10.

As mentioned in the Introduction, [12] also classify extreme measures on
infinite symmetric matrices
$\operatorname{Sym}(\mathbb{N},\mathbb{Q}_{p}):=\\{A\in\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p}):A^{T}=A\\}$
invariant under the action of $\operatorname{GL}_{\infty}(\mathbb{Z}_{p})$ by
$(g,A)\mapsto gAg^{T}$. The statement is more involved, essentially due to the
fact that the $\operatorname{GL}_{n}(\mathbb{Z}_{p})$-orbits on
$\operatorname{Sym}(n,\mathbb{Q}_{p})$ are parametrized by their singular
numbers together with additional data, unlike the
$\operatorname{GL}_{n}(\mathbb{Z}_{p})\times\operatorname{GL}_{m}(\mathbb{Z}_{p})$-orbits
on $\operatorname{Mat}_{n\times m}(\mathbb{Z}_{p})$. To pursue a similar
strategy to our proof of Section 1.3 one would need an analogue of 5.2, i.e. a
result giving the distribution of the
$\operatorname{GL}_{n-1}(\mathbb{Z}_{p})$-orbit of an $(n-1)\times(n-1)$
corner of an $n\times n$ symmetric matrix drawn uniformly from a fixed
$\operatorname{GL}_{n}(\mathbb{Z}_{p})$-orbit. Given that the parametrization
of these orbits involves more data than the (extended) signature specifying
their singular numbers, it is not immediately clear how the answer would be
expressed in terms of Hall-Littlewood polynomials.

We do however expect a solution in terms of Hall-Littlewood polynomials to a
related problem which is coarser. The problem is to find the distribution of
just the singular numbers, rather than
$\operatorname{GL}_{n-1}(\mathbb{Z}_{p})$-orbits, of an $(n-1)\times(n-1)$
corner of a random element of $\operatorname{Sym}(n,\mathbb{Q}_{p})$ with
fixed singular numbers and $\operatorname{GL}_{n}(\mathbb{Z}_{p})$-invariant
distribution. The existence of such a result is suggested by a known
expression for the singular numbers of an $n\times n$ symmetric matrix with
iid (apart from the symmetry constraint) entries distributed by the additive
Haar measure on $\mathbb{Z}_{p}$. This distribution was computed in [13], and
shown to be equivalent to a measure coming from one of the so-called
Littlewood identities for Hall-Littlewood polynomials in [20]. It seems
natural that a solution to this problem could be augmented with the extra data
required to parametrize $\operatorname{GL}_{n}(\mathbb{Z}_{p})$-orbits,
answering the question of the previous paragraph. We have not attempted to
pursue this direction.

##  6 Ergodic decomposition of $p$-adic Hua measures

We now define a special family of measures on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$, the _$p$ -adic Hua
measures_, introduced in [31]. Their decomposition into the ergodic measures
$\tilde{E}_{\mu}$ of 19 was computed in [1]. We will rederive that result,
showing in the process that the $p$-adic Hua measures have a natural
interpretation in terms of measures on partitions derived from Hall-Littlewood
polynomials.

######  Definition 20.

For $\lambda\in\operatorname{\mathbb{GT}}_{n}$, we set

$\lambda^{+}:=(\max(\lambda_{1},0),\ldots,\max(\lambda_{n},0))\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}.$

######  Definition 21.

The $p$-adic Hua measure $\mathbb{M}_{n}^{(s)}$ on
$\operatorname{Mat}_{n\times n}(\mathbb{Q}_{p})$ is defined by

$d\mathbb{M}_{n}^{(s)}(A)=\frac{(p^{-1-s};p^{-1})_{n}^{2}}{(p^{-1-s};p^{-1})_{2n}}p^{|\operatorname{ESN}(A)^{+}|(-s-2n)}d\mu_{Haar}^{(n)}(A),$

where $\mu_{Haar}^{(n)}$ is the product over all $n^{2}$ matrix entries of the
additive Haar measure $\mu_{Haar}$ on $\mathbb{Q}_{p}$.

The following computation of the distribution of the singular numbers of
$\mathbb{M}_{n}^{(s)}$ is done in [1, Proposition 3.1], using 21 and results
of [29, Chapter V].

######  Proposition 6.1.

The pushforward of $\mathbb{M}_{n}^{(s)}$ under
$\operatorname{SN}:\operatorname{Mat}_{n\times
n}(\mathbb{Q}_{p})\to\operatorname{\overline{\mathbb{GT}}}_{n}$ is supported
on $\operatorname{\mathbb{GT}}_{n}$ and given by

$\left(\operatorname{SN}_{*}(\mathbb{M}_{n}^{(s)})\right)(\lambda)=\frac{(u;t)_{n}^{2}}{(u;t)_{2n}}u^{|\lambda^{+}|}t^{(2n-1)(|\lambda^{+}|-|\lambda|)+2n(\lambda)}\frac{(t;t)_{n}^{2}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\lambda)}},$

where as usual $t=1/p$, and $u=t^{1+s}$.

We may now prove the main result, which we recall. Note that by 5.5 the same
result holds with $E_{\mu}$ replaced by $\tilde{E}_{\mu}$, and it is the
latter version which was proven in [1].

*

###### Proof.

The $p$-adic Hua measure is uniquely determined by its projections to $n\times
n$ corners, and by extremality of the measures $E_{\mu}$ any decomposition
into a convex combination of them is unique. Hence it suffices to show that a
matrix $A$, distributed by the measure on
$\operatorname{Mat}_{\infty\times\infty}(\mathbb{Q}_{p})$ described by
RHS(1.5), has $n\times n$ corners given by the finite $p$-adic Hua measure
$\mathbb{M}_{n}^{(s)}$. By 6.1 and Section 1.3, it suffices to show

$\displaystyle\begin{split}&\sum_{\mu\in\mathbb{Y}}\frac{P_{\mu}(1,t,\ldots)Q_{\mu}(u,ut,\ldots)}{\Pi(1,\ldots;u,\ldots)}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}\frac{P_{\mu/\lambda}(t^{n},\ldots)P_{\lambda}(1,\ldots,t^{n-1})}{P_{\mu}(1,t,\ldots)}Q_{-\nu/-\lambda}(t,t^{2},\ldots)\\\
\times&\frac{P_{-\nu}(1,\ldots,t^{n-1})}{P_{-\lambda}(1,\ldots,t^{n-1})\Pi(1,\ldots,t^{n-1};t,\ldots)}=\frac{(u;t)_{n}^{2}}{(u;t)_{2n}}u^{|\nu^{+}|}t^{(2n-1)(|\nu^{+}|-|\nu|)+2n(\nu)}\frac{(t;t)_{n}^{2}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\nu)}}\end{split}$
(6.1)

The proof is a surprisingly long series of applications of the Cauchy
identity/branching rule and principal specialization formulas. We first cancel
the $P_{\mu}(1,\ldots)$ factors and apply the Cauchy identity (2.10) to the
sum over $\mu$ to obtain

$\frac{P_{-\nu}(1,\ldots,t^{n-1})}{\Pi(1,\ldots;u,\ldots)\Pi(1,\ldots,t^{n-1};t,\ldots)}\\\
\times\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}\frac{P_{\lambda}(1,\ldots,t^{n-1})}{P_{-\lambda}(1,\ldots,t^{n-1})}Q_{-\nu/-\lambda}(t,\ldots)Q_{\lambda/(0[n])}(u,\ldots)\Pi(t^{n},\ldots;u,\ldots).$
(6.2)

Using that

$\Pi(1,\ldots,t^{n-1};t,\ldots)=(t;t)_{n},$

and

$P_{-\nu}(1,\ldots,t^{n-1})=P_{\nu}(1,\ldots,t^{-(n-1)})=t^{-(n-1)|\nu|}P_{\nu}(1,\ldots,t^{n-1})$

and similarly for $\lambda$, (6.2) becomes

$\frac{(t;t)_{n}P_{\nu}(1,\ldots,t^{n-1})t^{(n-1)(|\lambda|-|\nu|)}}{\Pi(1,\ldots,t^{n-1};u,\ldots)}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}Q_{-\nu/-\lambda}(t,\ldots)Q_{\lambda/(0[n])}(u,\ldots).$ (6.3)

It follows from the explicit branching rule 5 and the principal specialization
formula 2.3 for $P$ that

$Q_{-\nu/-\lambda}(x)=Q_{\lambda/\nu}(x)\frac{t^{-n(\lambda)}P_{\lambda}(1,\ldots,t^{n-1})}{t^{-n(\nu)}P_{\nu}(1,\ldots,t^{n-1})}.$
(6.4)

By definition of skew $Q$ functions (6.4) immediately extends to

$Q_{-\nu/-\lambda}(x_{1},\ldots,x_{k})=Q_{\lambda/\nu}(x_{1},\ldots,x_{k})\frac{t^{-n(\lambda)}P_{\lambda}(1,\ldots,t^{n-1})}{t^{-n(\nu)}P_{\nu}(1,\ldots,t^{n-1})}$

for any $k$, hence to an equality of symmetric functions and hence specializes
to

$Q_{-\nu/-\lambda}(t^{n},\ldots)=Q_{\lambda/\nu}(t^{n},\ldots)\frac{t^{-n(\lambda)}P_{\lambda}(1,\ldots,t^{n-1})}{t^{-n(\nu)}P_{\nu}(1,\ldots,t^{n-1})}.$
(6.5)

By first absorbing the $t^{(n-1)(|\lambda|-|\nu|)}$ into $Q_{-\nu/-\lambda}$
in (6.3) and then substituting (6.5) and simplifying $Q_{\lambda/(0[n])}$ via
2.3, (6.3) becomes

$\displaystyle\begin{split}&\frac{(t;t)_{n}P_{\nu}(1,\ldots,t^{n-1})}{\Pi(1,\ldots,t^{n-1};u,\ldots)}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}Q_{\lambda/\nu}(t^{n},\ldots)\frac{t^{-n(\lambda)}P_{\lambda}(1,\ldots,t^{n-1})}{t^{-n(\nu)}P_{\nu}(1,\ldots,t^{n-1})}u^{|\lambda|}t^{n(\lambda)}\\\
&=\frac{(t;t)_{n}t^{n(\nu)}}{\Pi(1,\ldots,t^{n-1};u,\ldots)}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}Q_{\lambda/\nu}(t^{n},\ldots)P_{\lambda}(u,\ldots,ut^{n-1}).\end{split}$
(6.6)

At first glance, the sum on the RHS of (6.6) looks like the one in the Cauchy
identity (2.8), but there is a nontrivial difference: the sum is over only
nonnegative signatures. If $\nu\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$
itself, this poses no issue and the Cauchy identity applies directly, but in
general this is not the case.

Luckily, using the explicit formula in 3.3 we may relate the sum in (6.6) to
one to which the Cauchy identity applies. By slightly rearranging terms in
3.3, we have that for $\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq 0}$,

$\displaystyle\begin{split}Q_{\lambda/\nu}(t^{n},\ldots)&=\frac{t^{n\cdot(|\lambda|-|\nu|)}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\nu)}}\prod_{x\leq
0}(t^{1+n-\nu_{x}^{\prime}};t)_{m_{x}(\nu)}t^{\binom{n-\nu_{x}^{\prime}}{2}}\prod_{x>0}(t^{1+\lambda_{x}^{\prime}-\nu_{x}^{\prime}};t)_{m_{x}(\nu)}t^{\binom{\lambda_{x}^{\prime}-\nu_{x}^{\prime}}{2}}\\\
Q_{\lambda/\nu^{+}}(t^{n},\ldots)&=\frac{t^{n\cdot(|\lambda|-|\nu^{+}|)}}{\prod_{x>0}(t;t)_{m_{x}(\nu)}}\prod_{x>0}(t^{1+\lambda_{x}^{\prime}-\nu_{x}^{\prime}};t)_{m_{x}(\nu)}t^{\binom{\lambda_{x}^{\prime}-\nu_{x}^{\prime}}{2}}\end{split}$
(6.7)

where $\nu^{+}$ is the truncation as in 20. Since

$\prod_{x\leq
0}(t^{1+n-\nu_{x}^{\prime}};t)_{m_{x}(\nu)}=(t;t)_{|\\{i:\nu_{i}\leq
0\\}|}=(t;t)_{m_{0}(\nu^{+})},$

(6.7) implies that

$Q_{\lambda/\nu}(t^{n},\ldots)=t^{n\cdot(|\nu^{+}|-|\nu|)+\sum_{x\leq
0}\binom{n-\nu_{x}^{\prime}}{2}}\frac{(t;t)_{m_{0}(\nu^{+})}}{\prod_{x\leq
0}(t;t)_{m_{x}(\nu)}}Q_{\lambda/\nu^{+}}(t^{n},\ldots).$

Therefore

$\displaystyle\begin{split}&\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}Q_{\lambda/\nu}(t^{n},\ldots)P_{\lambda}(u,\ldots,ut^{n-1})\\\
&=t^{n\cdot(|\nu^{+}|-|\nu|)+\sum_{x\leq
0}\binom{n-\nu_{x}^{\prime}}{2}}\frac{(t;t)_{m_{0}(\nu^{+})}}{\prod_{x\leq
0}(t;t)_{m_{x}(\nu)}}\sum_{\lambda\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}}Q_{\lambda/\nu^{+}}(t^{n},\ldots)P_{\lambda}(u,\ldots,ut^{n-1})\\\
&=t^{n\cdot(|\nu^{+}|-|\nu|)+\sum_{x\leq
0}\binom{n-\nu_{x}^{\prime}}{2}}\frac{(t;t)_{m_{0}(\nu^{+})}}{\prod_{x\leq
0}(t;t)_{m_{x}(\nu)}}\Pi(t^{n},\ldots;u,\ldots,ut^{n-1})P_{\nu^{+}}(u,\ldots,ut^{n-1})\\\
&=\frac{(t;t)_{n}}{(ut^{n};t)_{n}}\frac{u^{|\nu^{+}|}t^{n\cdot(|\nu^{+}|-|\nu|)+\sum_{x\leq
0}\binom{n-\nu_{x}^{\prime}}{2}+n(\nu^{+})}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\nu)}}\end{split}$
(6.8)

by applying (2.8) and 2.3. It is an elementary check from the definitions that

$n\cdot(|\nu^{+}|-|\nu|)+\sum_{x\leq
0}\binom{n-\nu_{x}^{\prime}}{2}+n(\nu^{+})=(2n-1)(|\nu^{+}|-|\nu|)+n(\nu).$
(6.9)

Substituting (6.9) into (6.8) and the result into (6.6) yields

$\displaystyle\begin{split}&(t;t)_{n}(u;t)_{n}t^{n(\nu)}\frac{(t;t)_{n}}{(ut^{n};t)_{n}}\frac{u^{|\nu^{+}|}t^{(2n-1)(|\nu^{+}|-|\nu|)+n(\nu)}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\nu)}}\\\
&=\frac{(u;t)_{n}^{2}}{(u;t)_{2n}}u^{|\nu^{+}|}t^{(2n-1)(|\nu^{+}|-|\nu|)+2n(\nu)}\frac{(t;t)_{n}^{2}}{\prod_{x\in\mathbb{Z}}(t;t)_{m_{x}(\nu)}},\end{split}$
(6.10)

which is the formula in 6.1, completing the proof. ∎

In some sense, the interpretation of the measures $M_{n}^{(s)}$ which we have
given here explains their special nature and gives a natural non-historical
route to their discovery. Let us suppose that one knew only 5.2 and Section
1.3, and wished to look for family of measures on $\operatorname{Mat}_{n\times
n}(\mathbb{Q}_{p})$ which are consistent under taking corners. Any measure on
the boundary yields such a family (and vice versa), but only for very nice
measures on the boundary do we expect the resulting measure on corners to have
any reasonable description. Because the cotransition probabilities feature
principal specializations, the natural candidate for this measure on the
boundary is a Hall-Littlewood measure with two principal specializations
$u_{1},u_{1}t,\ldots$ and $u_{2},u_{2}t,\ldots$. Indeed, the above
combinatorics would break down entirely for other Hall-Littlewood measures.
This leaves one free parameter because one may divide one specialization and
multiply the other by any positive real number without changing the measure,
and this free parameter is exactly the one in the $p$-adic Hua measure.

In another direction we note that, if one did not already know the result of
[1], the above considerations could help guess it. Since known natural
measures on finite $p$-adic matrices have singular numbers distributed by
Hall-Littlewood measures by [39, Theorem 1.3 and Corollary 1.4], and the
ergodic decomposition of a measure on infinite matrices is the analogue of the
distribution of singular numbers of a finite matrix, it is natural to search
for the ergodic decomposition within the space of Hall-Littlewood measures. As
mentioned above, essentially the only Hall-Littlewood measures with nice
explicit densities are those with principal specializations, of finite or
infinite length. If one were of finite length, say $N$, then it is a
straightforward consequence of Section 1.3 that at most $N$ singular numbers
of any corner are nonzero, which contradicts 6.1. Hence if the ergodic
decomposition is according to a well-behaved (principally specialized) Hall-
Littlewood measure, both specializations must be infinite, and this leads
exactly to the one-parameter family of Hall-Littlewood measures which do
indeed appear.

##  7 Products of finite $p$-adic random matrices

In this section, we prove the exact formula 1.1 for singular numbers of
products of Haar matrices. Recall that when $n\leq m$ and
$A\in\operatorname{Mat}_{n\times m}(\mathbb{Z}_{p})$ is nonsingular, the image
$\text{Im}(A)\subset\mathbb{Z}_{p}^{n}$ of the map
$A:\mathbb{Z}_{p}^{m}\to\mathbb{Z}_{p}^{n}$ is a $\mathbb{Z}_{p}$-submodule
and the cokernel $\operatorname{coker}(A):=\mathbb{Z}_{p}^{n}/\text{Im}(A)$ is
a finite abelian $p$-group, given by

$\operatorname{coker}(A)\cong\bigoplus_{i=1}^{n}\mathbb{Z}/p^{\lambda_{i}}\mathbb{Z}$

where $-\lambda=\operatorname{SN}(A)$. Most literature on $p$-adic random
matrices takes this perspective of random abelian $p$-groups, see the
references in the Introduction.

To relate cokernels/singular numbers of matrix products to Hall-Littlewood
combinatorics, we quote a special case of [39, Corollary 3.4], which states
that (negative444Let us reiterate that the sign convention on singular numbers
here is opposite from the one in [39] from which the above was taken, so 7.1
differs from the statement in [39, Corollary 3.4] by a sign.) singular numbers
of matrix products are distributed as a Hall-Littlewood process (defined
earlier in (2.12)). In what follows we use the notation $a[k]$ for variables
repeated $k$ times.

######  Proposition 7.1.

Let $t=1/p$, fix $n\geq 1$, and for $1\leq i\leq k$ let $A_{i}$ have iid
entries distributed by the additive Haar measure on $\mathbb{Z}_{p}$. Then for
$\lambda^{(1)},\ldots,\lambda^{(k)}\in\operatorname{\mathbb{GT}}_{n}^{\geq
0}$,

$\Pr(\operatorname{SN}(A_{i}\cdots A_{1})=-\lambda^{(i)}\text{ for all
}i=1,\ldots,k)=\frac{P_{\lambda^{(k)}}(1,\ldots,t^{n-1})\prod_{i=1}^{k}Q_{\lambda^{(i)}/\lambda^{(i-1)}}(t,t^{2},\ldots)}{\Pi(1,\ldots,t^{n-1};t[k],t^{2}[k],\ldots)}$

where we take $\lambda^{(0)}=(0[n])$.

We will deduce 1.1 from 7.1 together with the following, which uses results of
Section 3 to write an explicit formula for Hall-Littlewood process dynamics.

######  Proposition 7.2.

For $n\geq 1$ and $\lambda,\nu\in\operatorname{\mathbb{GT}}_{n}$, we have

$\frac{Q_{\nu/\lambda}(u,ut,\ldots)P_{\nu}(1,\ldots,t^{n-1})}{P_{\lambda}(1,\ldots,t^{n-1})\Pi(1,\ldots,t^{n-1};u,ut,\ldots)}=(u;t)_{n}u^{|\nu|-|\lambda|}t^{n(\nu)-n(\lambda)+n(\nu/\lambda)}\prod_{x\in\mathbb{Z}}\begin{bmatrix}\nu_{x}^{\prime}-\lambda_{x+1}^{\prime}\\\
\nu_{x}^{\prime}-\nu_{x+1}^{\prime}\end{bmatrix}_{t}.$ (7.1)

###### Proof.

It follows from the definition in (2.9) and telescoping that

$\frac{1}{\Pi(1,\ldots,t^{n-1};u,ut,\ldots)}=(u;t)_{n}.$

Combining 3.3 with 2.3 yields

$\frac{Q_{\nu/\lambda}(u,ut,\ldots)P_{\nu}(1,\ldots,t^{n-1})}{P_{\lambda}(1,\ldots,t^{n-1})}=u^{|\nu|-|\lambda|}t^{n(\nu/\lambda)+n(\nu)-n(\lambda)}\prod_{x\in\mathbb{Z}}\frac{(t^{1+\nu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}}{(t;t)_{m_{x}(\nu)}}$

Noting that

$\prod_{x\in\mathbb{Z}}\frac{(t^{1+\nu_{x}^{\prime}-\lambda_{x}^{\prime}};t)_{m_{x}(\lambda)}}{(t;t)_{m_{x}(\nu)}}=\prod_{x\in\mathbb{Z}}\begin{bmatrix}\nu_{x}^{\prime}-\lambda_{x+1}^{\prime}\\\
\nu_{x}^{\prime}-\nu_{x+1}^{\prime}\end{bmatrix}_{t}$

completes the proof. ∎

###### Proof of 1.1.

Follows immediately by combining 7.1 and 7.2 with $u=t$. ∎