# Cokernels of random matrix products and flag Cohen–Lenstra heuristic

Yifeng Huang  Dept. of Mathematics, University of British Columbia
<EMAIL_ADDRESS>
###### Abstract.

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective
flag of ${\mathbb{Z}}_{p}$-modules modeled by $\mathrm{cok}(M_{1}\cdots
M_{k})\twoheadrightarrow\dots\twoheadrightarrow\mathrm{cok}(M_{1})$ for
independent random matrices
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ satisfies the
Cohen–Lenstra heuristic. We answer the question affirmatively when
$M_{1},\dots,M_{k}$ follow the Haar measure, and our proof demonstrates how
classical ideas in Cohen–Lenstra heuristic adapt naturally to the flag
setting. We also prove an analogue for non-square matrices.

##  1\. Introduction

We start by introducing our notation for surjective flags of
${\mathbb{Z}}_{p}$-modules, which is our main subject of investigation.

###  1.1. Notation and terminology

Fix $k\in{\mathbb{Z}}_{\geq 1}$ and a prime $p$. Let
$\mathbf{Mod}_{{\mathbb{Z}}_{p}}$ denote the category of finitely generated
${\mathbb{Z}}_{p}$-modules. We refer to a diagram of surjections
$G_{k}\overset{\phi_{k-1}}{\twoheadrightarrow}\dots\overset{\phi_{1}}{\twoheadrightarrow}G_{1}(\overset{\phi_{0}}{\twoheadrightarrow}G_{0}:=0)$
in $\mathbf{Mod}_{{\mathbb{Z}}_{p}}$ as a $k$-surjective flag of (finitely
generated) ${\mathbb{Z}}_{p}$-modules. We denote by
$\mathbf{Fl}_{k}=\mathbf{Fl}_{k}({\mathbb{Z}}_{p})$ the set of $k$-surjective
flags of ${\mathbb{Z}}_{p}$-modules up to isomorphism. For
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1})\in\mathbf{Fl}_{k}$, let $\operatorname{Aut}(\mathbf{G})$ denote the
automorphism of the flag $\mathbf{G}$. See also [NVP22, Def. 18, 19].

Let $n\in{\mathbb{Z}}_{\geq 1}$ and let
$\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ denote the set of $n\times n$
matrices over ${\mathbb{Z}}_{p}$. For
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$, consider the
flag

$\mathrm{cok}(M_{1}\cdots
M_{k})\twoheadrightarrow\dots\twoheadrightarrow\mathrm{cok}(M_{1}),$ (1.1)

where the surjection $\mathrm{cok}(M_{1}\dots
M_{i+1})={\mathbb{Z}}_{p}^{n}/\operatorname{im}(M_{1}\dots
M_{i+1})\twoheadrightarrow\mathrm{cok}(M_{1}\dots
M_{i})={\mathbb{Z}}_{p}^{n}/\operatorname{im}(M_{1}\dots M_{i})$ is induced by
the inclusion $\operatorname{im}(M_{1}\dots
M_{i+1})\subseteq\operatorname{im}(M_{1}\dots M_{i})$. We denote this flag by
$\mathbf{cok}(M_{1},\dots,M_{k})$.

Since $\operatorname{Mat}_{n}({\mathbb{Z}}_{p})\simeq{\mathbb{Z}}_{p}^{n^{2}}$
is a compact topological group, there is a unique probability Haar measure on
$\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$. We refer to a random element of
$\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ following the Haar measure as a
Haar-random matrix in $\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$.

###  1.2. Background

When $M_{1},\dots,M_{k}$ are random matrices in
$\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$, we get a probability measure on
$\mathbf{Fl}_{k}$ modeled by $\mathbf{cok}(M_{1},\dots,M_{k})$. In [NVP22],
Nguyen and Van Peski initiated the investigation of
$\mathbf{cok}(M_{1},\dots,M_{k})$ by studying the joint distribution of
$\mathrm{cok}(M_{1}),\dots,\allowbreak\mathrm{cok}(M_{1}\cdots M_{k})$ as
${\mathbb{Z}}_{p}$-modules. They proved a universality result in the sense of
Wood [Woo19], namely, if the $kn^{2}$ entries from $M_{1},\dots,M_{k}$ are
independent and each is not too concentrated mod $p$, then as $n\to\infty$,
the limiting joint distribution of
$\mathrm{cok}(M_{1}),\dots,\mathrm{cok}(M_{1}\cdots M_{k})$ is insensitive to
the exact distributions of these $kn^{2}$ entries. The limiting joint
distribution is also explicitly determined.

Since the datum $(\mathrm{cok}(M_{1}),\dots,\mathrm{cok}(M_{1}\cdots M_{k}))$
is just $\mathbf{cok}(M_{1},\dots,M_{k})$ forgetting the surjections in
between, it is natural to expect that the above results are explained by a
universal distribution on the level of
$\mathbf{cok}(M_{1},\dots,M_{k})\in\mathbf{Fl}_{k}$. Nguyen and Van Peski
[NVP22, §10] asked if the above results lift to
$\mathbf{cok}(M_{1},\dots,M_{k})$; they defined the conjectured limiting
distribution in [NVP22, Thm. 1.3].

###  1.3. Our result

We answer their question affirmatively when $M_{1},\dots,M_{k}$ distribute
independently and follow the Haar measure. In this case, we also obtain the
exact distribution of $\mathbf{cok}(M_{1},\dots,M_{k})$ for each fixed $n$.
For $G\in\mathbf{Mod}({\mathbb{Z}}_{p})$, let
$r(G):=\dim_{{{\mathbb{F}}_{p}}}G/pG$ be the rank of $G$.

######  Theorem 1.1.

Fix $k\in{\mathbb{Z}}_{\geq 1}$. Let
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ be independent
and Haar-random, and fix
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1})\in\mathbf{Fl}_{k}$ such that $\lvert G_{k}\rvert<\infty$. Then for
$n\geq r(G_{k})$,

$\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathbf{cok}(M_{1},\dots,M_{k})\simeq\mathbf{G})=\frac{1}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\left\lparen\prod_{i=n-r(G_{k})+1}^{n}\\!\\!(1-p^{-i})\right\rparen\left\lparen\prod_{i=1}^{n}(1-p^{-i})\right\rparen^{k}.$
(1.2)

In particular, when $n\to\infty$,

$\lim_{n\to\infty}\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathbf{cok}(M_{1},\dots,M_{k})\simeq\mathbf{G})=\frac{1}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\left\lparen\prod_{i=1}^{\infty}(1-p^{-i})\right\rparen^{k}.$
(1.3)

######  Remark.

It is clear that if $\lvert G_{k}\rvert=\infty$ or $n<r(G_{k})$, then the
probability in (1.2) is zero: $G_{k}=\mathrm{cok}(M_{1}\cdots M_{k})$ must
have rank at most $n$, and $\lvert G_{k}\rvert=\infty$ only happens when at
least one of $\det(M_{1}),\dots,\allowbreak\det(M_{k})$ is zero, which happens
with probability zero.

The flag $\mathbf{cok}(M_{1},\dots,M_{k})$ is probably the finest datum one
could get from a chain of matrices $(M_{1},\dots,M_{k})$.111One could think of
$(M_{1},\dots,M_{k})$ as a chain of linear maps
${\mathbb{Z}}_{p}^{n}\overset{M_{k}}{\to}\dots\overset{M_{1}}{\to}{\mathbb{Z}}_{p}^{n}$.
For example, $\mathrm{cok}(M_{i}M_{i+1}\dots M_{j})$ for $1\leq i\leq j\leq k$
is isomorphic to the kernel of $G_{j}\twoheadrightarrow G_{i-1}$, a
concatenation of several surjections from $\mathbf{cok}(M_{1},\dots,M_{k})$.
The distribution of $\mathbf{cok}(M_{1},\dots,M_{k})$ thus encodes the joint
distribution of all $\mathrm{cok}(M_{i}M_{i+1}\dots M_{j})$. We note that the
joint distribution of certain subsets of these cokernels have natural
connections to Hall algebras (see [NVP22, p. 46]).

An important purpose of the paper is to demonstrate that some classical ideas
to study the Cohen–Lenstra heuristic [CL84, FW89] adapt to the (apparently
highly refined) flag setting nicely. This point will be evident once we set up
the language in §2.1. The ease to work with flags will be showcased in our
proof of Theorem 1.1; for example, no knowledge about
$\lvert\operatorname{Aut}(\mathbf{G})\rvert$ is required.222Indeed when $k\geq
2$, we do not even have a combinatorial classification of $\mathbf{G}$ [NVP22,
p. 44, Rmk. 7], let alone a general formula for
$\lvert\operatorname{Aut}(\mathbf{G})\rvert$. However, it is known that
certain summations involving $1/\lvert\operatorname{Aut}(\mathbf{G})\rvert$
are expressed in Hall–Littlewood polynomials [NVP22, §5–7,10]. In light of its
simplicity, we conjecture that a suitable combination of our method and the
general machinery of Sawin and Wood [SW22] would yield a universality version
of Theorem 1.1.

###  1.4. Further applications

A slight modification of the proof of Theorem 1.1 implies an analogue for non-
square matrices.

######  Theorem 1.2.

Fix $k\in{\mathbb{Z}}_{\geq 1}$, and $u_{1},\dots,u_{k}\in{\mathbb{Z}}_{\geq
0}$. For $1\leq i\leq k$, let
$M_{i}\in\operatorname{Mat}_{(n+u_{i-1})\times(n+u_{i})}$, where $u_{0}:=0$,
and assume $M_{1},\dots,M_{k}$ are independent and Haar-random. Fix
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{0}=0)\in\mathbf{Fl}_{k}$. Then

$\operatorname*{Prob}_{M_{1},\dots,M_{k}}(\mathbf{cok}(M_{1},\dots,M_{k})\simeq\mathbf{G})=\frac{\prod_{j=1}^{k}\left\lparen\frac{\lvert
G_{j}\rvert}{\lvert
G_{j-1}\rvert}\right\rparen^{-u_{j}}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\left\lparen\prod_{i=n-r(G_{k})+1}^{n}\\!\\!(1-p^{-i})\right\rparen\left\lparen\prod_{j=1}^{k}\prod_{i=1}^{n}(1-p^{-i-u_{j}})\right\rparen$
(1.4)

if $\lvert G_{k}\rvert<\infty$ and $r(G_{k})\leq n$, and zero otherwise. In
particular, when $n\to\infty$,

$\lim_{n\to\infty}\operatorname*{Prob}_{M_{1},\dots,M_{k}}(\mathbf{cok}(M_{1},\dots,M_{k})\simeq\mathbf{G})=\frac{\prod_{j=1}^{k}\left\lparen\frac{\lvert
G_{j}\rvert}{\lvert
G_{j-1}\rvert}\right\rparen^{-u_{j}}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\prod_{j=1}^{k}\prod_{i=1}^{\infty}(1-p^{-i-u_{j}})$
(1.5)

if $\lvert G_{k}\rvert<\infty$, and zero otherwise.

######  Remark.

It is less obvious _a priori_ why $\lvert\mathrm{cok}(M_{1}\cdots
M_{k})\rvert<\infty$ with probability one, since we cannot use determinants
anymore. This will be a consequence of our proof.

The non-flag case (i.e., $k=1$) was proved in [Woo19], together with the
universality result. We conjecture that (1.5) holds if $M_{1},\dots,M_{k}$ are
independent and $\varepsilon$-balanced in the definition of [Woo19].

As an application of the non-square analogue, we construct a $k$-parameter
family of deformations of the “Cohen–Lenstra probability measure” in (1.2).
For any flag $\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{0}=0)\in\mathbf{Fl}_{k}$ such that $\lvert G_{k}\rvert<\infty$, we let
$n_{1}(\mathbf{G}),\dots,n_{k}(\mathbf{G})\in{\mathbb{Z}}_{\geq 0}$ be defined
by

$p^{n_{i}(\mathbf{G})}:=\lvert G_{i}\rvert/\lvert G_{i-1}\rvert.$ (1.6)

######  Definition 1.3.

For $k,n\in{\mathbb{Z}}_{\geq 1},t_{1},\dots,t_{k}\in[0,p)$, we define a
measure $P_{n,(t_{1},\dots,t_{k})}$ on $\mathbf{Fl}_{k}$ by

$P_{n,(t_{1},\dots,t_{k})}(\mathbf{G}):=\frac{\prod_{j=1}^{k}t_{j}^{n_{j}(\mathbf{G})}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\left\lparen\prod_{i=n-r(G_{k})+1}^{n}\\!\\!(1-p^{-i})\right\rparen\left\lparen\prod_{j=1}^{k}\prod_{i=1}^{n}(1-p^{-i}t_{j})\right\rparen$
(1.7)

if $\lvert G_{k}\rvert<\infty$ and $r(G_{k})\leq n$, and zero otherwise.
Similarly, define

$P_{\infty,(t_{1},\dots,t_{k})}(\mathbf{G}):=\frac{\prod_{j=1}^{k}t_{j}^{n_{j}(\mathbf{G})}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\left\lparen\prod_{j=1}^{k}\prod_{i=1}^{\infty}(1-p^{-i}t_{j})\right\rparen$
(1.8)

if $\lvert G_{k}\rvert<\infty$, and zero otherwise.

When $t_{i}=1$, the measure (1.7) reduces to (1.2); when
$t_{i}=p^{-u_{i}},u_{i}\in{\mathbb{Z}}_{\geq 0}$, the measure (1.7) reduces to
(1.4). This means that in these cases, (1.7) comes from a random matrix model.
A brief argument in §5 implies:

######  Corollary 1.4.

$P_{n,(t_{1},\dots,t_{k})}$ and $P_{\infty,(t_{1},\dots,t_{k})}$ are
probability measures on $\mathbf{Fl}_{k}$.

When $k=1$, Definition 1.3 is precisely the measure considered by Fulman and
Kaplan [FK19]. The joint distribution of $(G_{1},\dots,G_{k})$ for
$\mathbf{G}$ distributed according to (1.7) is expressible in terms of
Hall–Littlewood polynomials and can be extracted from [VP21], see Proposition
5.1.

Part of the content of Corollary 1.4 is that the normalizing constant in
Definition 1.3 is correct. For example, $P_{\infty,(t_{1},\dots,t_{k})}$ being
a probability measure is equivalent to the formal identity

$\sum_{\begin{subarray}{c}\mathbf{G}\\\ \lvert
G_{k}\rvert<\infty\end{subarray}}\frac{\prod_{j=1}^{k}t_{j}^{n_{j}(\mathbf{G})}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}=\prod_{j=1}^{k}\prod_{i=1}^{\infty}\frac{1}{1-p^{-i}t_{j}}\in{\mathbb{C}}[[t_{1},\dots,t_{k}]].$
(1.9)

Its function field analogue naturally connects to matrices in a parabolic
subalgebra; see §5.1.

Deformations of the Cohen–Lenstra measure are already interesting without
motivations from random matrix models; see [DJ14, FK19] and [NVP22, §7] for
the intrinsic study of some probability measures of this type, and note the
ubiquity of Hall–Littlewood polynomials. But furthermore, such deformations
also arise as predicted distributions of arithmetic objects: for example
[CL84], if $p$ is odd and we take $k=1,u=u_{1},G=\mathbf{G}=G_{1}$ in (1.5),
then the resulting probability mass function, proportional to $1/(\lvert
G\rvert^{u}\lvert\operatorname{Aut}(G)\rvert)$, predicts the distribution of
the $p$-part of the class group of a random quadratic extension of
${\mathbb{Q}}$ ($u=0$ for imaginary, $u=1$ for real). We propose the following
question to conclude the introduction.

######  Question 1.5.

Are there arithmetic settings that naturally produce a flag of finite abelian
groups? Is the distribution of its $p$-part predicted by one of the
distributions above?

### Acknowledgements

The author thanks Roger Van Peski for fruitful discussions and comments on
earlier drafts.

##  2\. Preliminaries

###  2.1. Dictionary of flags

Given $n,k\in{\mathbb{Z}}_{\geq 1}$ and a surjective flag
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1})\in\mathbf{Fl}_{k}$, a surjection from ${\mathbb{Z}}_{p}^{n}$ to
$\mathbf{G}$ simply refers to a surjective ${\mathbb{Z}}_{p}$-linear map
$f_{k}:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow G_{k}$. Any such surjection
$f_{k}$ induces a chain of surjections

${\mathbb{Z}}_{p}^{n}\overset{f_{k}}{\twoheadrightarrow}G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1}\twoheadrightarrow G_{0}=0,$ (2.1)

which induces a surjection $f_{i}:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow
G_{i}$ for $0\leq i\leq k$ by composition. We can equivalently think of a
surjection $f:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow\mathbf{G}$ as a
collection $(f_{i}:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow G_{i})_{i}$, but
keeping in mind that $f_{k}$ determines the rest.

By a $k$-injective flag in ${\mathbb{Z}}_{p}^{n}$, we mean a tower of
${\mathbb{Z}}_{p}$-submodules of ${\mathbb{Z}}_{p}^{n}$:

$F_{k}\subseteq F_{k-1}\subseteq\dots\subseteq F_{1}\subseteq
F_{0}:={\mathbb{Z}}_{p}^{n}.$ (2.2)

We denote by $\mathcal{F}l_{k}({\mathbb{Z}}_{p}^{n})$ the set of $k$-injective
flags in ${\mathbb{Z}}_{p}^{n}$.

There is a one-to-one correspondence between
$\mathcal{F}l_{k}({\mathbb{Z}}_{p}^{n})$ and the set of $k$-surjective flags
$\mathbf{G}$ together with a surjection
${\mathbb{Z}}_{p}^{n}\twoheadrightarrow\mathbf{G}$. To
$\mathcal{F}=(F_{k}\subseteq\dots\subseteq F_{0}={\mathbb{Z}}_{p}^{n})$, we
associate
${\mathbb{Z}}_{p}^{n}\twoheadrightarrow\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{0}=0)$, where $G_{i}={\mathbb{Z}}_{p}^{n}/F_{i}$ for $0\leq i\leq k$ and
the maps are the natural quotient maps. Conversely, given
$f:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow\mathbf{G}$, we recover $\mathcal{F}$
by $F_{i}=\ker(f_{i}:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow G_{i})$ for $0\leq
i\leq k$. We introduce the natural notation
$\mathbf{G}={\mathbb{Z}}_{p}^{n}/\mathcal{F}$ and
$\mathcal{F}=\mathit{ker}(f:{\mathbb{Z}}_{p}^{n}\twoheadrightarrow\mathbf{G})$.

Given matrices $M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$,
we define an injective flag by $F_{i}=\operatorname{im}(M_{1}\dots
M_{i})\subseteq{\mathbb{Z}}_{p}^{n}$ for $1\leq i\leq k$. We denote this flag
by $\mathit{im}(M_{1},\dots,M_{k})$. We canonically have
$\mathbf{cok}(M_{1},\dots,M_{k})\simeq{\mathbb{Z}}_{p}^{n}/\mathit{im}(M_{1},\dots,M_{k})$.

##  3\. Proof of Theorem 1.1

Fix $\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1})\in\mathbf{Fl}_{k}$ with $\lvert G_{k}\rvert<\infty$. Let $r:=r(G_{k})$,
and fix $n\geq r$. We separate the major steps of proving Theorem 1.1 into the
following lemmas.

######  Lemma 3.1.

The number of $k$-injective flags $\mathcal{F}$ in ${\mathbb{Z}}_{p}^{n}$ such
that ${\mathbb{Z}}_{p}^{n}/\mathcal{F}\simeq\mathbf{G}$ is

$\frac{\lvert
G_{k}\rvert^{n}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\prod_{i=n-r(G_{k})+1}^{n}(1-p^{-i}).$
(3.1)

###### Proof.

Fix a copy of $\mathbf{G}$, and let
$\mathrm{Surj}({\mathbb{Z}}_{p}^{n},\mathbf{G})$ denote the set of surjections
from ${\mathbb{Z}}_{p}^{n}$ to $\mathbf{G}$, which is nothing but the set of
surjections ${\mathbb{Z}}_{p}^{n}\twoheadrightarrow G_{k}$. By Nakayama’s
lemma,

$\lvert\mathrm{Surj}({\mathbb{Z}}_{p}^{n},\mathbf{G})\rvert=\lvert
G_{k}\rvert^{n}\prod_{i=n-r+1}^{n}(1-p^{-i}).$ (3.2)

Let $\operatorname{Aut}(\mathbf{G})$ act on
$\mathrm{Surj}({\mathbb{Z}}_{p}^{n},\mathbf{G})$ by composition. As usual, the
action is free: if $\sigma=(\sigma_{i})\in\operatorname{Aut}(\mathbf{G})$ with
$\sigma_{i}\in\operatorname{Aut}(G_{i})$ is such that $\sigma_{i}\circ
f_{i}=f_{i}$ for all $i$, then since $f_{i}$ is surjective, we must have
$\sigma_{i}=\mathrm{id}$. As a consequence, the orbit space has cardinality
given by

$\left\lvert\frac{\mathrm{Surj}({\mathbb{Z}}_{p}^{n},\mathbf{G})}{\operatorname{Aut}(\mathbf{G})}\right\rvert=\frac{\left\lvert\mathrm{Surj}({\mathbb{Z}}_{p}^{n},\mathbf{G})\right\rvert}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}=\frac{\lvert
G_{k}\rvert^{n}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\prod_{i=n-r+1}^{n}(1-p^{-i}).$
(3.3)

Finally, it is easy to verify that the orbit space above is in a canonical
bijection with the set of $k$-injective flags $\mathcal{F}$ with
${\mathbb{Z}}_{p}^{n}/\mathcal{F}\simeq\mathbf{G}$. The conclusion then
follows. ∎

######  Lemma 3.2.

Fix a $k$-injective flag $\mathcal{F}$ such that
${\mathbb{Z}}_{p}^{n}/\mathcal{F}\simeq\mathbf{G}$. Then if
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ are independent
and Haar-random, then

$\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathit{im}(M_{1},\dots,M_{k})=\mathcal{F})=\lvert
G_{k}\rvert^{-n}\left\lparen\prod_{i=1}^{n}(1-p^{-i})\right\rparen^{k}.$ (3.4)

###### Proof.

Let $\mathcal{F}=(F_{k}\subseteq\dots\subseteq
F_{1}\subseteq{\mathbb{Z}}_{p}^{n})$. Since $\lvert
G_{k}\rvert=\lvert{\mathbb{Z}}_{p}^{n}/F_{k}\rvert<\infty$, every $F_{i}$ is a
free module of rank $n$. We first pick $M_{1}$ with the condition that
$\operatorname{im}(M_{1})=F_{1}$. This means two things: (1)
$M_{1}:{\mathbb{Z}}_{p}^{n}\to{\mathbb{Z}}_{p}^{n}$ has image in $F_{1}$. (2)
The induced ${\mathbb{Z}}_{p}$-linear map $M_{1}:{\mathbb{Z}}_{p}^{n}\to
F_{1}$ is surjective.

The probability that $\operatorname{im}(M_{1})\subseteq F_{1}$ is the
probability that every column of $M_{1}$ lies in $F_{1}$. Hence
$\operatorname*{Prob}_{M_{1}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\operatorname{im}(M_{1})\subseteq
F_{1})$ is $\lvert{\mathbb{Z}}_{p}^{n}/F_{1}\rvert^{-n}=\lvert
G_{1}\rvert^{-n}$. By Nakayama’s lemma, since $F_{1}$ is of rank $n$, the
probability that a Haar-random linear map ${\mathbb{Z}}_{p}^{n}\to F_{1}$ be
surjective is $\prod_{i=1}^{n}(1-p^{-i})$. As a result,

$\operatorname*{Prob}_{M_{1}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\operatorname{im}(M_{1})=F_{1})=\lvert
G_{1}\rvert^{-n}\prod_{i=1}^{n}(1-p^{-i}).$ (3.5)

Now we fix $M_{1}$ and pick $M_{2}$ with the condition that
$\operatorname{im}(M_{1}M_{2})=F_{2}$. We note that
$M_{1}:{\mathbb{Z}}_{p}^{n}\to F_{1}$ is an isomorphism because it is a
surjective map between rank $n$ free modules over ${\mathbb{Z}}_{p}$.
Therefore, $M_{1}$ induces an isomorphism of flags from
$M_{1}^{-1}(F_{2})\subseteq{\mathbb{Z}}_{p}^{n}$ to $F_{2}\subseteq F_{1}$.
Thus, $\operatorname{im}(M_{1}M_{2})=F_{2}$ if and only if
$\operatorname{im}(M_{2})=M_{1}^{-1}(F_{2})\subseteq{\mathbb{Z}}_{p}^{n}$. By
the same argument as above, and the fact that
$\lvert{\mathbb{Z}}_{p}^{n}/M_{1}^{-1}(F_{2})\rvert=\lvert
F_{1}/F_{2}\rvert=\lvert G_{2}\rvert/\lvert G_{1}\rvert$, we get

$\operatorname*{Prob}_{M_{2}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}\left(\operatorname{im}(M_{1}M_{2})=F_{2}|\operatorname{im}(M_{1})=F_{1}\right)=\left\lparen\frac{\lvert
G_{2}\rvert}{\lvert G_{1}\rvert}\right\rparen^{-n}\prod_{i=1}^{n}(1-p^{-i}).$
(3.6)

Repeating the argument and multiplying all of the above probabilities
together, we conclude that

$\begin{aligned}
&\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathit{im}(M_{1},\dots,M_{k})=\mathcal{F})\\\
&=\lvert
G_{1}\rvert^{-n}\prod_{i=1}^{n}(1-p^{-i})\cdot\left\lparen\frac{\lvert
G_{2}\rvert}{\lvert
G_{1}\rvert}\right\rparen^{-n}\prod_{i=1}^{n}(1-p^{-i})\cdot\dots\cdot\left\lparen\frac{\lvert
G_{k}\rvert}{\lvert
G_{k-1}\rvert}\right\rparen^{-n}\prod_{i=1}^{n}(1-p^{-i})\\\ &=\lvert
G_{k}\rvert^{-n}\left\lparen\prod_{i=1}^{n}(1-p^{-i})\right\rparen^{k}.\end{aligned}\qed$
(3.7)

###### Proof of Theorem 1.1.

Since
$\mathbf{cok}(M_{1},\dots,M_{k})\simeq{\mathbb{Z}}_{p}^{n}/\mathit{im}(M_{1},\dots,M_{k})$,
we have

$\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathbf{cok}(M_{1},\dots,M_{k})\simeq\mathbf{G})=\sum_{\begin{subarray}{c}\mathcal{F}\in\mathcal{F}l_{k}({\mathbb{Z}}_{p}^{n})\\\
{\mathbb{Z}}_{p}^{n}/\mathcal{F}\simeq\mathbf{G}\end{subarray}}\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathit{im}(M_{1},\dots,M_{k})=\mathcal{F}).$
(3.8)

Since the probability in Lemma 3.2 depends only on $\mathbf{G}$ but not on
$\mathcal{F}$, the proof is complete by multiplying the results of Lemmas 3.1
and 3.2. ∎

##  4\. Proof of Theorem 1.2

We state and prove a convenient lemma first.

######  Lemma 4.1.

Let $A,B,C$ be free modules over ${\mathbb{Z}}_{p}$ of finite ranks $a,b,c$,
and suppose $f:B\twoheadrightarrow A$ is a surjective linear map. Then for a
Haar-random linear map $g:C\to B$, we have

$\operatorname*{Prob}_{g\in\operatorname{Hom}(C,B)}(\operatorname{im}(fg)=A)=\prod_{i=c-a+1}^{c}(1-p^{-i}).$
(4.1)

###### Proof.

Without loss of generality, we may assume
$A={\mathbb{Z}}_{p}^{a},B={\mathbb{Z}}_{p}^{b},C={\mathbb{Z}}_{p}^{c}$ and
$f:{\mathbb{Z}}_{p}^{b}\to{\mathbb{Z}}_{p}^{a}$ is the projection to the first
$a$ coordinates. Write $g\in\operatorname{Mat}_{b\times c}({\mathbb{Z}}_{p})$
as $g=\begin{bmatrix}g_{1}\\\ g_{2}\end{bmatrix}$, where
$g_{1}\in\operatorname{Mat}_{a\times c}({\mathbb{Z}}_{p})$ and
$g_{2}\in\operatorname{Mat}_{(b-a)\times c}({\mathbb{Z}}_{p})$. Then
$fg=g_{1}$. Thus, the probability that $fg$ be surjective is the probability
that $g_{1}$ be surjective, which is $\prod_{i=c-a+1}^{c}(1-p^{-i})$ by
Nakayama’s lemma. ∎

To prove Theorem 1.2, we follow the same argument as Theorem 1.1, except that
we need to prove a more general version of Lemma 3.2. Again, we fix
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{1})\in\mathbf{Fl}_{k}$, but we do not assume $\lvert G_{k}\rvert<\infty$.
Fix $\mathcal{F}\in\mathcal{F}l_{k}({\mathbb{Z}}_{p}^{n})$ such that
${\mathbb{Z}}_{p}^{n}/\mathcal{F}\simeq\mathbf{G}$. For $1\leq i\leq k$, let
$M_{i}\in\operatorname{Mat}_{(n+u_{i-1})\times(n+u_{i})}$, where $u_{0}:=0$,
and assume $M_{1},\dots,M_{k}$ are independent and Haar-random.

######  Lemma 4.2.

In the setting above, we have

$\operatorname*{Prob}_{M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})}(\mathit{im}(M_{1},\dots,M_{k})=\mathcal{F})=\lvert
G_{k}\rvert^{-n}\prod_{j=1}^{k}\left\lparen\frac{\lvert G_{j}\rvert}{\lvert
G_{j-1}\rvert}\right\rparen^{-u_{j}}\cdot\prod_{j=1}^{k}\prod_{i=1}^{n}(1-p^{-i-u_{j}})$
(4.2)

if $\lvert G_{k}\rvert<\infty$, and zero otherwise.

###### Proof.

Let $\mathcal{F}=(F_{k}\subseteq\dots\subseteq
F_{1}\subseteq{\mathbb{Z}}_{p}^{n})$. We first pick
$M_{1}\in\operatorname{Mat}_{n\times(n+u_{1})}$ with
$\operatorname{im}(M_{1})=F_{1}$. If $\lvert G_{1}\rvert=\infty$, then
$\lvert{\mathbb{Z}}_{p}^{n}/F_{1}\rvert=\infty$, so the probability that each
column of $M_{1}$ be in $F_{1}$ is zero. Therefore, we may assume $\lvert
G_{1}\rvert<\infty$ from now on. As a result, $F_{1}$ is free of rank $n$. By
the similar argument in the proof of Lemma 3.2, we get

$\operatorname*{Prob}_{M_{1}\in\operatorname{Mat}_{n\times(n+u_{1})}({\mathbb{Z}}_{p})}(\operatorname{im}(M_{1})=F_{1})=\lvert
G_{1}\rvert^{-(n+u_{1})}\prod_{i=u_{1}+1}^{n+u_{1}}(1-p^{-i})=\lvert
G_{1}\rvert^{-(n+u_{1})}\prod_{i=1}^{n}(1-p^{-i-u_{1}}).$ (4.3)

Now we fix $M_{1}$ and pick
$M_{2}\in\operatorname{Mat}_{(n+u_{1})\times(n+u_{2})}$ with the condition
that $\operatorname{im}(M_{1}M_{2})=F_{2}$, which is equivalent to
$\operatorname{im}(M_{2})\subseteq M_{1}^{-1}(F_{2})$. The third isomorphism
theorem applied to the surjection $M_{1}:{\mathbb{Z}}_{p}^{n+u_{1}}\to F_{1}$
gives an isomorphism ${\mathbb{Z}}_{p}^{n+u_{1}}/M_{1}^{-1}(F_{2})\simeq
F_{1}/F_{2}$, so
$\lvert{\mathbb{Z}}_{p}^{n+u_{1}}/M_{1}^{-1}(F_{2})\rvert=\lvert
F_{1}/F_{2}\rvert=\lvert G_{2}\rvert/\lvert G_{1}\rvert$. Thus,

$\operatorname*{Prob}_{M_{2}\in\operatorname{Mat}_{(n+u_{1})\times(n+u_{2})}}(\operatorname{im}(M_{1}M_{2})\subseteq
F_{2}|\operatorname{im}(M_{1})=F_{1})=\left\lparen\frac{\lvert
G_{2}\rvert}{\lvert G_{1}\rvert}\right\rparen^{-(n+u_{2})}$ (4.4)

if $\lvert G_{2}\rvert<\infty$, and zero otherwise. So again, we may assume
$\lvert G_{2}\rvert<\infty$ from now on.

We now find the probability that $\operatorname{im}(M_{1}M_{2})=F_{2}$
conditioned on $\operatorname{im}(M_{1}M_{2})\subseteq F_{2}$, so $M_{2}$ is a
Haar-random linear map in
$\operatorname{Hom}({\mathbb{Z}}_{p}^{n+u_{2}},M_{1}^{-1}(F_{2}))$. Note that
we are in the setting of Lemma 4.1 with $A=F_{2}$, $B=M_{1}^{-1}(F_{2})$,
$C={\mathbb{Z}}_{p}^{n+u_{2}}$, $f=M_{1}:M_{1}^{-1}(F_{2})\twoheadrightarrow
F_{2}$, and $g=M_{2}:{\mathbb{Z}}_{p}^{n+u_{2}}\to M_{1}^{-1}(F_{2})$. The
condition that $\operatorname{im}(M_{1}M_{2})=F_{2}$ is equivalent to
$\operatorname{im}(fg)=A$. Since
$\lvert{\mathbb{Z}}_{p}^{n}/F_{2}\rvert=\lvert G_{2}\rvert<\infty$, $A$ is
free of rank $n$. Since $\lvert{\mathbb{Z}}_{p}^{n+u_{1}}/B\rvert=\lvert
G_{2}\rvert/\lvert G_{1}\rvert<\infty$, $B$ is free of rank $n+u_{1}$. By
Lemma 4.1 with $a=n$ and $c=n+u_{2}$, we get

$\operatorname*{Prob}_{M_{2}\in\operatorname{Mat}_{(n+u_{1})\times(n+u_{2})}}(\operatorname{im}(M_{1}M_{2})=F_{2}|\operatorname{im}(M_{1})=F_{1},\operatorname{im}(M_{1}M_{2})\subseteq
F_{2})=\prod_{i=u_{2}+1}^{n+u_{2}}(1-p^{-i}).$ (4.5)

Combined with (4.4), we get

$\operatorname*{Prob}_{M_{2}\in\operatorname{Mat}_{(n+u_{1})\times(n+u_{2})}}(\operatorname{im}(M_{1}M_{2})=F_{2}|\operatorname{im}(M_{1})=F_{1})=\left\lparen\frac{\lvert
G_{2}\rvert}{\lvert
G_{1}\rvert}\right\rparen^{-(n+u_{2})}\prod_{i=1}^{n}(1-p^{-i-u_{2}}).$ (4.6)

Repeating the argument inductively and multiplying the probabilities in (4.3),
(4.6), and so on, the desired formula (4.2) follows, along with the fact that
each $G_{i}$ must be finite in order for the probability to be nonzero. ∎

###### Proof of Theorem 1.2.

Multiply the results of Lemma 3.1 and Lemma 4.2. ∎

##  5\. Probability measures on flags

Here we prove Corollary 1.4 as an immediate consequence of Theorem 1.2.

###### Proof of Corollary 1.4.

It is clear that $P_{n,(t_{1},\dots,t_{k})}$ is a nonnegative measure, so it
suffices to show that
$\sum_{\mathbf{G}\in\mathbf{Fl}_{k}}P_{n,(t_{1},\dots,t_{k})}(\mathbf{G})=1$.
We note that the measure in (1.4) is precisely
$P_{n,(p^{-u_{1}},\dots,p^{-u_{k}})}$. In particular, Theorem 1.2 implies that
$P_{n,(p^{-u_{1}},\dots,p^{-u_{k}})}$ is a probability measure for every
$u_{1},\dots,u_{k}\in{\mathbb{Z}}_{\geq 0}$, so the formal power series
$\sum_{\mathbf{G}\in\mathbf{Fl}_{k}}P_{n,(t_{1},\dots,t_{k})}(\mathbf{G})$ in
$t_{1},\dots,t_{k}$ must be 1, completing the proof.

The case of $P_{\infty,(t_{1},\dots,t_{k})}$ is similar. ∎

Write $\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow G_{1})$. We
compute the joint distribution of $(G_{1},\dots,G_{k})$ if $\mathbf{G}$ is
distributed according to (1.7). For partitions
$\lambda^{(1)},\dots,\lambda^{(k)}$, we write
$\mathbf{G}\sim(\lambda^{(1)},\dots,\lambda^{(k)})$ if the type of $G_{i}$ is
$\lambda^{(i)}$.

######  Proposition 5.1.

Fixing partitions $\lambda^{(1)},\dots,\lambda^{(k)}$. For
$P_{n,(t_{1},\dots,t_{k})}(\mathbf{G})$ as in (1.7), the quantity

$\sum_{\mathbf{G}\sim(\lambda^{(1)},\dots,\lambda^{(k)})}P_{n,(t_{1},\dots,t_{k})}(\mathbf{G})$
(5.1)

is given by [VP21, p. 19, Prop. 2.6] with $q=0$, $a^{(i)}_{j}=t_{i}p^{-j}$,
$\mathbf{b}=(1,p^{-1},\dots,p^{-(n-1)})$, and $t=1/p$ in their notation.

###### Proof.

If $t_{i}=1$, by Theorem 1.1, the quantity in question is the probability that
$\mathrm{cok}(M_{1}\dots M_{i})$ is of type $\lambda^{(i)}$ for all $i$, where
$M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p})$ are independent
and Haar-random. This is given by [VP21, p. 27, Cor. 3.4] with $N_{i}=\infty$
in their notation.

To go from the $t_{i}=1$ case to the general case, we notice that if we fix
$\varnothing=\lambda^{(0)},\lambda^{(1)},\dots,\lambda^{(k)}$ and letting
$n_{i}=\lvert\lambda^{(i)}\rvert-\lvert\lambda^{(i-1)}\rvert$ for $1\leq i\leq
k$, then from (1.7), we have

$\frac{\sum_{\mathbf{G}\sim(\lambda^{(1)},\dots,\lambda^{(k)})}P_{n,(t_{1},\dots,t_{k})}(\mathbf{G})}{\sum_{\mathbf{G}\sim(\lambda^{(1)},\dots,\lambda^{(k)})}P_{n,(1,\dots,1)}(\mathbf{G})}=\prod_{j=1}^{k}\left\lparen
t_{j}^{n_{j}}\prod_{i=1}^{n}\frac{1-p^{-i}t_{j}}{1-p^{-i}}\right\rparen.$
(5.2)

Combining this with the $t_{i}=1$ case above finally gives the desired
formula. ∎

###  5.1. Relation to matrices over finite fields

We give a direct proof of Proposition 5.2, a function field analogue of (1.9),
by establishing (5.6) that connects it to counting matrices over finite
fields. Let ${\mathbb{F}}_{q}$ be the finite field with $q$ elements, and let
$R$ denote the power series ring ${\mathbb{F}}_{q}[[T]]$. We can similarly
define $\mathbf{Fl}_{k}(R)$ to be the set of $k$-surjective flags of
$R$-modules. Given $1\leq i\leq k$ and any flag
$\mathbf{G}=(G_{k}\twoheadrightarrow\dots\twoheadrightarrow
G_{0}=0)\in\mathbf{Fl}_{k}(R)$ such that
$\dim_{{\mathbb{F}}_{q}}G_{k}<\infty$, let
$n_{i}(\mathbf{G}):=\dim_{{\mathbb{F}}_{q}}G_{i}-\dim_{{\mathbb{F}}_{q}}G_{i-1}$.
Define the flag Cohen–Lenstra series of $R$ as

$\widehat{Z}_{R}(t_{1},\dots,t_{k}):=\sum_{\begin{subarray}{c}\mathbf{G}\in\mathbf{Fl}_{k}(R)\\\
\dim
G_{k}<\infty\end{subarray}}\frac{\prod_{j=1}^{k}t_{j}^{n_{j}(\mathbf{G})}}{\lvert\operatorname{Aut}(\mathbf{G})\rvert}\in{\mathbb{Q}}[[t_{1},\dots,t_{k}]].$
(5.3)

When $k=1$, this construction is precisely the Cohen–Lenstra series defined in
[Hua23].

######  Proposition 5.2.

We have

$\widehat{Z}_{{\mathbb{F}}_{q}[[T]]}(t_{1},\dots,t_{k})=\prod_{j=1}^{k}\prod_{i=1}^{\infty}\frac{1}{1-q^{-i}t_{j}}.$
(5.4)

###### Proof.

To specify a $k$-surjective flag $\mathbf{G}$ of
${\mathbb{F}}_{q}[[T]]$-modules with given $n_{i}(\mathbf{G})=n_{i}$, it
suffices to specify the $k$-surjective flag of underlying
${\mathbb{F}}_{q}$-vector spaces

$V_{k}\overset{\phi_{k-1}}{\twoheadrightarrow}\dots\overset{\phi_{1}}{\twoheadrightarrow}V_{1}\twoheadrightarrow
V_{0}=0,$ (5.5)

together with ${\mathbb{F}}_{q}[[T]]$-module structures on $V_{i}$ that are
compatible with the flag. Up to isomorphism of flags of
${\mathbb{F}}_{q}$-vector spaces, we may assume
$V_{i}={\mathbb{F}}_{q}^{n_{1}+\dots+n_{i}}$, and the surjection $\phi_{i}$ is
the projection to the first $n_{1}+\dots+n_{i}$ coordinates. Then the
compatible ${\mathbb{F}}_{q}[[T]]$-module structures on $V_{i}$ are determined
by a nilpotent endomorphism $M$ on $V_{k}$ (as the multiplication by $T$ map),
such that $M$ factors through an endomorphism on $V_{i}$ for each $i$. If
$W_{i}:=\ker(V_{k}\twoheadrightarrow V_{i})$ for $0\leq i\leq k$, then this
simply means $MW_{i}\subseteq W_{i}$ for $0\leq i\leq k$. Since $W_{i}$ is the
span of the last $n_{i+1}+\dots+n_{k}$ basis vectors in
$V_{k}={\mathbb{F}}_{q}^{n_{1}+\dots+n_{k}}$, this happens if and only if $M$
is block-lower-triangular with respect to the block sizes $n_{1},\dots,n_{k}$
in both rows and columns. We denote the set of such block-lower-triangular
matrices by $\operatorname{Mat}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$.

Let $\operatorname{GL}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$,
$\mathrm{Nilp}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$ be the set of invertible
resp. nilpotent matrices in
$\operatorname{Mat}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$. By the discussion
above, $k$-surjective flags of ${\mathbb{F}}_{q}[[T]]$-modules with a given
dimension vector $(n_{i})$ are parametrized by
$\mathrm{Nilp}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$. The group
$\operatorname{GL}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$ acts on
$\mathrm{Nilp}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$ by conjugation. The
orbits correspond to isomorphism classes of flags of
${\mathbb{F}}_{q}[[T]]$-modules, and the stabilizers correspond to
automorphisms of flags of ${\mathbb{F}}_{q}[[T]]$-modules. By a standard
argument involving the orbit-stabilizer theorem, we get

$\widehat{Z}_{{\mathbb{F}}_{q}[[T]]}(t_{1},\dots,t_{k})=\sum_{n_{1},\dots,n_{k}\geq
0}\frac{\lvert\mathrm{Nilp}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})\rvert}{\lvert\operatorname{GL}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})\rvert}t_{1}^{n_{1}}\dots
t_{k}^{n_{k}}.$ (5.6)

Since a matrix in $\operatorname{Mat}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$
is invertible resp. nilpotent if and only if every diagonal block is
invertible resp. nilpotent, the generating series simplifies to

$\widehat{Z}_{{\mathbb{F}}_{q}[[T]]}(t_{1},\dots,t_{k})=\prod_{j=1}^{k}\sum_{n_{j}\geq
0}\frac{\lvert\mathrm{Nilp}_{n_{j}}({\mathbb{F}}_{q})\rvert}{\lvert\operatorname{GL}_{n_{j}}({\mathbb{F}}_{q})\rvert}t_{j}^{n_{j}}.$
(5.7)

But it is well-known that the sum equals to
$\prod_{i=1}^{\infty}\frac{1}{1-q^{-i}t_{j}}$; for instance, we may use the
theorem $\lvert\mathrm{Nilp}_{n}({\mathbb{F}}_{q})\rvert=q^{n^{2}-n}$ of Fine
and Herstein [FH58], and then apply an identity of Euler [And98, Eq. (2.2.5)].
This completes the proof. ∎

######  Remark.

Of course, $\operatorname{Mat}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$ and
$\operatorname{GL}_{n_{1},\dots,n_{k}}({\mathbb{F}}_{q})$ are nothing but the
parabolic subalgebra and the parabolic subgroup of a suitable partial flag
variety of type A. As is made apparent by Equation (5.6), the construction
(5.3) fits naturally into the study of commuting varieties of parabolic
subalgebras; we refer the readers to [BE16] for a geometric aspect of such
research.

## References

  * [And98] George E. Andrews.  The theory of partitions.  Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998.  Reprint of the 1976 original.
  * [BE16] Michaël Bulois and Laurent Evain.  Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras.  J. Lie Theory, 26(2):497–533, 2016.
  * [CL84] H. Cohen and H. W. Lenstra, Jr.  Heuristics on class groups of number fields.  In Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume 1068 of Lecture Notes in Math., pages 33–62. Springer, Berlin, 1984.
  * [DJ14] Christophe Delaunay and Frédéric Jouhet.  $p^{\ell}$-torsion points in finite abelian groups and combinatorial identities.  Adv. Math., 258:13–45, 2014.
  * [FH58] N. J. Fine and I. N. Herstein.  The probability that a matrix be nilpotent.  Illinois J. Math., 2:499–504, 1958.
  * [FK19] Jason Fulman and Nathan Kaplan.  Random partitions and Cohen-Lenstra heuristics.  Ann. Comb., 23(2):295–315, 2019.
  * [FW89] Eduardo Friedman and Lawrence C. Washington.  On the distribution of divisor class groups of curves over a finite field.  In Théorie des nombres (Quebec, PQ, 1987), pages 227–239. de Gruyter, Berlin, 1989.
  * [Hua23] Yifeng Huang.  Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity.  J. Algebra, 619:26–50, 2023.
  * [NVP22] Hoi H. Nguyen and Roger Van Peski.  Universality for cokernels of random matrix products.  Preprint. https://arxiv.org/abs/2209.14957v2, 2022.
  * [SW22] Will Sawin and Melanie Matchett Wood.  The moment problem for random objects in a category.  Preprint. https://arxiv.org/abs/2210.06279, 2022.
  * [VP21] Roger Van Peski.  Limits and fluctuations of $p$-adic random matrix products.  Selecta Math. (N.S.), 27(5):Paper No. 98, 71, 2021.
  * [Woo19] Melanie Matchett Wood.  Random integral matrices and the Cohen-Lenstra heuristics.  Amer. J. Math., 141(2):383–398, 2019.


# The first non-static inhomogeneous exact solution in $f(T,B)$ gravity

Sebastián Nájera<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U.,
A.P. 70-543, México D.F. 04510, México.  Aram Aguilar
<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad
Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F.
04510, México.  Celia Escamilla-Rivera<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Circuito Exterior C.U., A.P. 70-543, México D.F. 04510, México.  Roberto A.
Sussman<EMAIL_ADDRESS>Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543,
México D.F. 04510, México.

###### Abstract

We examine in this paper the possibility of finding exact solutions for
Teleparallel Gravity (TG) of the type of spherically symmetric Lemaître-
Tolman-Bondi (LTB) dust models. We apply to the LTB metric, as obtained from
the Schwarzschild solution in General Relativity, the formalism of
Teleparallel Gravity in its extension to $f(T,B)$ models. An exact LTB
solution is obtained that is compatible with a specific $f(T,B)$ model that
seems to be appropriate to fit observations when applied to standard spatially
flat Robertson-Walker geometry.

Introduction.- Cosmology has benefited in the last decades from an enormous
increase of new observational data whose proper interpretation is still a
challenging endeavor, not only because the fundamental nature of the dark
sector is still an open research topic, but also from existing tensions around
the values of $H_{0}$ and $S_{8}$ and anomalies in early-time observations,
all of which deserve further investigation DiValentino:2020vhf ; di2021realm .

There is currently a wide variety of dark energy proposals to explain the
large scale accelerated cosmic expansion under the framework of General
Relativity (GR) under gauge invariant cosmological perturbations on a
Friedmann-Lemaître-Robertson-Walker (FLRW) background. An accelerated cosmic
expansion can be obtained by assuming a positive $\Lambda$, extra dark fluids
with negative pressure, scalar fields or modifications of General Relativity
(GR), among others Clifton:2011jh ; ishak2019modified . More recently,
extensions of Teleparallel Gravity (TG) Cai:2015emx ; Hohmann:2019nat  have
also been considered to explain cosmic acceleration, as well as to alleviate
the $H_{0}$ tension, all this from the geometry of the theory itself without
involving an extra fluid or $\Lambda$ Briffa:2020qli  and in compatibility
with current observations Nunes:2018xbm ; Escamilla-Rivera:2019ulu ;
Cai:2019bdh .

TG is a theory of gravity in which the curvature-based description of gravity,
used in GR, is enhanced by torsion, which involves replacing the Levi-Civita
connection with its Weitzenböck connection analog. The name of teleparallelism
or distant parallelism comes from the geometrical description of vectors in
the Weitzenböck space-time Pereira:2013qza , where torsion can change the
direction of such vectors koivisto . TG is a gauge theory for the translations
group article , so it is possible to represent the gravitational field with a
translational gauge potential that appears within the non-trivial part of the
tetrad field (and in fact this keeps torsion different from zero).

In this paper we study the compatibility between TG theories and the standard
spherically symmetric Lemaîitre-Tolman-Bondi (LTB) models, an important class
of exact inhomogeneous solutions of GR with a dust source Krasinski ;
plebanski , though the LTB metric also admits non-trivial solutions with non-
zero pressure Sussman:PRD . These solutions are very useful as toy models to
study structure formation lake2000gravitational ; sussman2009quasilocal ;
deshingkar2001gravitational ; bolejko2010structures , as they can also be
conceived as exact non-linear perturbations of a FLRW background, reducing in
the linear regime to standard cosmological perturbations in the isochronous
comoving gauge sussman2015spherical ; Sussman:Invar ; sussman2008quasi ;
sussman2010new .

While there is an extensive literature on LTB models in GR, in TG theories
only a few articles have considered them to examine the possibility to
describe dark energy dynamics without introducing a cosmological constant
Cai:2015emx  and in the discussion on the problem of energy-momentum
localization in TG salti2005energy . In this context, our interest is to
examine how the known properties of LTB solutions can be affected by torsion
acting as a force in a possible LTB solution of TG theories.

From Schwarzschild to LTB solutions: the GR approach.- Consider the
Schwarzschild spacetime in standard static coordinates111We will consider
geometric units ($c=G=1$) along the rest of the paper.

$ds^{2}=-\left(1-\frac{2M_{0}}{r}\right)dt^{2}+\left(1-\frac{2M_{0}}{Y}\right)^{-1}dY^{2}+Y^{2}d\Omega^{2},$
(1)

where $M_{0}$ is the Schwarzchild mass. Its corresponding radial timelike
geodesics follow from:

$\dot{Y}^{2}=\frac{2M_{0}}{Y}-k,\quad\dot{t}=\frac{\sqrt{1-\kappa}}{1-2M_{0}/Y},\quad
k\equiv\frac{k_{0}^{2}}{4}-1,$ (2)

where $k_{0}$ is the conserved energy in each geodesic. From Eq. (2) we can
construct a coordinate system $(\tau,r)$ for the Schwarzschild spacetime based
on the world lines of observers following radial geodesics plebanski  with
their proper time $\tau$ as time coordinate and $r$ acting as a continuous
parameter that marks each geodesic. Since the binding energy $k$ continuously
varies along the geodesic congruence in the $(\tau,r)$ plane, we have
necessarily: $k=k(r)$ and $Y=Y(\tau,r),\,\,t=t(\tau,r)$. Under the coordinate
transformation $(t,Y)\to(\tau,r)$ defined by Eq. (2) the Schwarzschild static
metric takes the following time dependent form:

$ds^{2}=-d\tau^{2}+\frac{Y^{\prime 2}}{1-k}\,dr^{2}+Y^{2}\,d\Omega^{2},$ (3)

where $k=k(r)$ and
$\mathcal{A}^{\prime}\equiv\left[\partial\mathcal{A}/\partial r\right]_{\tau}$
for all functions $\mathcal{A}$, with $Y(\tau,r)$ satisfying

$\displaystyle\dot{Y}^{2}=\frac{2M_{0}}{Y}+k,$ (4)
$\displaystyle\quad\tau-\tau_{B}(r)=\int_{\bar{Y}=0}^{Y}{\frac{\sqrt{\bar{Y}}d\bar{Y}}{\sqrt{2M_{0}+k(r)\,\bar{Y}}}},$

where $\dot{\mathcal{A}}\equiv
u^{a}\mathcal{A}_{,a}=\left[\partial\mathcal{A}/\partial\tau\right]_{r}$ for
every function $\mathcal{A}$ and $\tau_{B}(r)$ an integration constant marking
the value of $\tau$ for which $Y=0$ (i.e. the geodesics fall or arise from the
singularity at different values of $\tau$ for each $r$). These coordinates
make it easy to appreciate that the Schwarzschild radius $Y_{s}=2M_{0}$ is a
regular locus.

The Schwarzschild mass can be represented in the spherical coordinates of Eq.
(3) as an integrated distribution $M_{0}=\int_{V}{\rho\,d^{3}x}$ with
$\rho=M_{0}\delta^{3}(0)/(r^{2}\sin\theta)$. Substituting the distributional
density for a continuous density (equivalent to consider
$M_{0}\,\rightarrow\,M=M(r)$) and assuming the $4$-velocity
$u^{a}=\left[dx^{a}/d\tau\right]_{r}=\delta^{a}_{\tau}$, for observers
comoving along the geodesics, the substitution of Eqs. (3)–(4) in Einstein
equations, $G_{ab}=8\pi\Theta_{ab}$ yields

$\Theta^{ab}=\rho\,u^{a}\,u^{b}=\rho\,\delta^{a}_{\tau}\,\delta^{b}_{\tau},\quad
8\pi\rho=\frac{2M^{\prime}}{Y^{2}\,Y^{\prime}},\quad u^{a}=\delta^{a}_{\tau},$
(5)

which characterize LTB models with a dust source (in fact, Eq. (3) is already
the the LTB metric representation of the Schwarzschild solution). It is worth
noting that Eq. (4) need not hold in modified gravity theories, for example in
f(R) theories (see sussman2017lemaitre ). Nevertheless, we will consider
henceforth this evolution equation for the area radius $Y$, Eq. (4) in the
derivation of Schwarzschild-LTB models in TG gravity.

Teleparallel Gravity solutions.- As a gauge theory, the gravitational
interaction in TG is modeled as a force and the particles trajectories are not
geodesic but force equations, with the torsion tensor playing the role of
force krvsvsak2019teleparallel . Classically speaking, it is completely
equivalent to describe gravitation with a pseudo-Riemannian with signature
$(-,+,+,+)$ (GR) or a Weitzenböck (TG) space-time structure Pereira4 .
Moreover, by a convenient choice of the TG Lagrangian, GR and TG are equal up
to a boundary term, which means that they produce identical dynamical
equations Bahamonde:2019shr . The relation between the two theories is really
important in the sense that one should always look for certain TG models that
in a particular limit can return GR, this is appropriate if we want to
describe gravity at the level where GR is useful. In this equivalence, the
geodesic equation is analog to the Lorentz force equation of electrodynamics.
TG is based on the tetrad formalism the tetrads or tetrad fields are
considered the dynamical variables of the system. In contrast to GR in which
the metric takes the role of dynamical variable celia3 . In the Weitzenböck
spacetime structure, a Weitzenböck connection
$\Gamma_{\enspace\mu\nu}^{\sigma}$ is defined as

$\Gamma_{\enspace\mu\nu}^{\sigma}:=E_{A}^{\enspace\sigma}\partial_{\mu}e_{\enspace\nu}^{A}+E_{A}^{\enspace\sigma}\omega_{\enspace
B\mu}^{A}e_{\enspace\nu}^{B},$ (6)

where $e_{\enspace\mu}^{A}$ is the tetrad field, $E_{A}^{\enspace\sigma}$ the
transpose tetrad field defined by
$E_{A}^{\enspace\mu}e_{\enspace\nu}^{A}=\delta^{\mu}_{\nu}$ and
$\omega_{\enspace b\mu}^{A}$ is the spin connection. Space-time indices are
denoted by greek letters while capital letters stand for tangent space
indices. In all this work, quantities with/without a white circle upon them
are calculated with the Levi-Civita/Weitzenböck connection.

The spin connection serves to retain the invariance of the TG field equations
under local Lorentz transformations Golovnev:2017dox . Hence, for any
particular tetrad, the spin connection balances this freedom with different
inertial contributions, and can vanish for particular frame choices.
Henceforth we consider the Weitzenböck gauge $\omega_{\enspace b\mu}^{a}=0$.
One must ensure to choose a tetrad which obeys this gauge, otherwise one could
restrict the theory. This problem is also known as a choice of good and bad
tetrads Hohmann:2019nat ; krvsvsak2019teleparallel .

The action in TG theories has several cases, the first and easiest is called
Teleparallel Equivalent of General Relativity (TEGR). This is very important
because the variation of such action with respect to the tetrad field results
in completely equivalent to GR dynamical equations Bahamonde:2019jkf . The
Lagrangian density in this case takes the form $-T+B$. Raising the TEGR action
to its $f(T,B)$ gravity extension results in

$\mathcal{S}=\int d^{4}x\bigg{[}\dfrac{1}{16\pi}f(T,B)+\mathcal{L}_{\rm
m}\bigg{]}e,$ (7)

where $e$ is the determinant of the tetrad field
$e=\det(e^{A}_{\enskip\mu})=\sqrt{-g}$. $T$ is the torsion scalar and $B$ is
the boundary term denoted, respectively, by

$T:=S_{A}^{\enspace\mu\nu}T_{\enspace\mu\nu}^{A},\quad
B=\frac{2}{e}\partial_{\mu}\Big{(}e\tensor{T}{{}^{\nu}_{\nu}{}^{\mu}}\Big{)},$
(8)

where we use the superpotential and the torsion tensor defined as

$S_{A}^{\enspace\mu\nu}:=\Gamma_{\enspace\mu\nu}^{\sigma}-\mathring{\Gamma}_{\enspace\mu\nu}^{\sigma}-E_{A}^{\enspace\nu}T_{\quad\alpha}^{\alpha\mu}+E_{A}^{\enspace\mu}T_{\quad\alpha}^{\alpha\nu}.$
(9) $T_{\enspace\mu\nu}^{\sigma}:=2\Gamma_{\enspace[\nu\mu]}^{\sigma}.$ (10)

$\mathcal{L}_{\rm m}$ represents the Lagrangian for matter and $f=f(T,B)$ is
an arbitrary function of $T$ and $B$.

This generalization of TG provides a richer class of models and is quite
powerful as it allows to compare TG with $f(\mathring{R})$ models. $f(T,B)$
gravity has also been well-studied sebas2 ; capo2 ; andro ; gabriel2 ; sebas4
; matthew ; Bahamonde:2016grb ; Bahamonde:2020lsm  as a possible extension to
TEGR. Taking a variation of Eq. (7) with respect to the tetrad field leads to

$\displaystyle
2\delta_{\nu}^{\lambda}\mathring{\Box}f_{B}-2\mathring{\nabla}^{\lambda}\mathring{\nabla}_{\nu}f_{B}+Bf_{B}\delta_{\nu}^{\lambda}+4\partial_{\mu}(f_{B}+f_{T})S_{\nu}{}^{\mu\lambda}+4e^{-1}e^{A}{}_{\nu}\partial_{\mu}(eS_{A}{}^{\mu\lambda})f_{T}-4f_{T}T^{\sigma}{}_{\mu\nu}S_{\sigma}{}^{\lambda\mu}-f\delta_{\nu}^{\lambda}=16\pi\Theta_{\nu}^{\lambda},$
(11)

where $\mathring{\Box}=\mathring{\nabla}^{\delta}\mathring{\nabla}_{\delta}$,
$\Theta^{\alpha\mu}$ is the energy-momentum tensor, and $f_{T}$ and $f_{B}$
refer to the derivative of $f$ with respect to the torsion scalar and boundary
term, respectively.

From Schwarzschild to LTB: The Teleparallel approach.- From Eq. (3) we use a
tetrad which is compatible with the Weitzenböck gauge Hohmann:2019nat ;
bahamonde2020general

$e^{a}{}_{\mu}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\frac{Y^{\prime}\cos\phi\sin\theta}{\sqrt{1-k}}&Y\cos\phi\cos\theta&-Y\sin\phi\sin\theta\\\
0&\frac{Y^{\prime}\sin\phi\sin\theta}{\sqrt{1-k}}&Y\sin\phi\cos\theta&Y\cos\phi\sin\theta\\\
0&\frac{Y^{\prime}\cos\theta}{\sqrt{1-k}}&-Y\sin\theta&0\\\
\end{array}\right)\,.$ (12)

With this choice of tetrad and considering a dust source, $\Theta_{ab}=\rho
u_{a}u_{b}$ with $\rho$ the matter–energy density, the non-diagonal components
of Eq. (11) read

$\displaystyle\dot{f}^{\prime}_{B}=\frac{\dot{Y}^{\prime}f^{\prime}_{B}}{Y^{\prime}}+\frac{2\dot{Y}(f^{\prime}_{B}+f^{\prime}_{T})}{Y},$
(13)
$\displaystyle(-1+\sqrt{1-k})(\dot{f}_{B}+\dot{f}_{T})Y^{\prime}=\sqrt{1-k}(f^{\prime}_{B}+f^{\prime}_{T})\dot{Y},$
(14)

while the independent diagonal components are

$\displaystyle-\frac{\bigg{(}\Big{[}]4\sqrt{1-k}-4\Big{]}Y^{\prime}-2k^{\prime}Y+8M^{\prime}\bigg{)}f_{T}}{Y^{\prime}Y^{2}}-\frac{2[1-k]f^{\prime\prime}_{B}}{Y^{\prime
2}}+\frac{\Big{(}2(1-k)YY^{\prime\prime}+k^{\prime}YY^{\prime}-4\sqrt{1-k}Y^{\prime
2}\Big{)}f^{\prime}_{B}}{YY^{\prime 3}}$
$\displaystyle-\frac{\Big{(}-4\dot{Y}YY^{\prime}-2Y^{2}\dot{Y}^{\prime}\Big{)}\dot{f}_{B}}{Y^{\prime}Y^{2}}+\frac{4\sqrt{1-k}\Big{(}\sqrt{1-k}-1\Big{)}f^{\prime}_{T}}{Y^{\prime}Y}-Bf_{B}+f=16\pi\rho,$
(15)
$\displaystyle-\frac{\bigg{(}\Big{(}4\sqrt{1-k}-4\Big{)}Y^{\prime}-2k^{\prime}Y+8M^{\prime}\bigg{)}f_{T}}{Y^{\prime}Y^{2}}-\frac{2(1-k)f^{\prime\prime}_{B}}{Y^{\prime
2}}+\frac{\Big{(}2(1-k)YY^{\prime\prime}+k^{\prime}YY^{\prime}-4\sqrt{1-k}Y^{\prime
2}\Big{)}f^{\prime}_{B}}{YY^{\prime 3}}$ $\displaystyle
Bf_{B}+\frac{\bigg{(}\Big{(}4\sqrt{1-k}-4\Big{)}Y^{\prime}-2k^{\prime}Y+4M^{\prime}\bigg{)}f_{T}}{Y^{2}Y^{\prime}}-2\ddot{f}_{B}+\frac{4(1-k)f^{\prime}_{B}}{YY^{\prime}}-f+\frac{4\dot{Y}\dot{f}_{T}}{Y}=0,$
(16) $\displaystyle\frac{2(1-k)f^{\prime\prime}_{B}}{Y^{\prime
2}}+Bf_{B}+\Bigg{(}\frac{4\sqrt{1-k}-4}{Y^{2}}+\frac{-2k^{\prime}Y+4M^{\prime}}{Y^{2}Y^{\prime}}\Bigg{)}f_{T}-2\ddot{f}_{B}-\frac{\Big{(}2(1-k)YY^{\prime\prime}+k^{\prime}YY^{\prime}-2\sqrt{1-k}Y^{\prime
2}\Big{)}f^{\prime}_{B}}{YY^{\prime 3}}$
$\displaystyle-f-\frac{2\sqrt{1-k}\Big{(}\sqrt{1-k}-1\Big{)}f^{\prime}_{T}}{Y^{\prime}Y}+\frac{2(\dot{Y}Y^{\prime}+Y\dot{Y}^{\prime})\dot{f}_{T}}{Y^{\prime}Y}=0.$
(17)

To find an exact solution to Eq. (11) we consider the following ansatz

$\dot{f}_{T}=-\dot{f}_{B},\quad f^{\prime}_{T}=-f^{\prime}_{B},$ (18)

in Eqs. (13)-(17). Eq. (13) integrates to

$f^{\prime}_{B}=AY^{\prime}.$ (19)

We consider now the spatially flat subcase $k=0$ of Eq. (3) ($k$ is
proportional to the 3-dimensional Ricci scalar of constant $\tau$ slices). For
this case Eq. (4) has a closed solution

$Y=\left(\frac{9}{2}M(t-t_{b}(r))^{2}\right)^{1/3},$ (20)

which reduces the complexity of Eqs. (15)-(17), through a functional form for
$f$ is still needed to obtain an exact TG solution. In particular, the Mixed
Power Law Model (MPLM)

$f(T,B)=\mathcal{F}B^{n}T^{m},$ (21)

where $\mathcal{F}$, $n$ and $m$ are free parameters, applied to FLRW metric
seems to be a convenient choice complying with viable cosmological scenarios
under TG, leading to a power law solution for the scale factor
Bahamonde:2016grb  thus allowing for an evolution that either mimics a phantom
energy, or preserves quintessence behavior at high redshifts Escamilla-
Rivera:2019ulu , admitting as well a continuous transition between
cosmological epochs.

Substituting Eq. (21) and Eq. (20) into Eq. (19), we obtain

$\displaystyle AY^{3/2}M^{\prime
3}+C_{nm}\left(\frac{MM^{\prime}}{-M\sqrt{2}Y^{\frac{3}{2}}\sqrt{M}t^{\prime}_{B}+\frac{1}{3}M^{\prime}Y^{3}}\right)^{m+n}(m+n-1)n\mathcal{F}\Xi=3AM^{3/2}t^{\prime}_{B}\sqrt{2}M^{\prime
2},$ (22)

where $C_{nm}=2^{2m+n-1}3^{n+1}$, and

$\Xi=t^{\prime}_{B}M^{2}\sqrt{Y}M^{\prime}t^{\prime}_{B}+\frac{1}{3}Y^{2}\sqrt{2}\Big{(}t^{\prime}_{B}M^{\frac{3}{2}}M^{\prime\prime}-t^{\prime\prime}_{B}M^{\frac{3}{2}}M^{\prime}-2M^{\frac{1}{2}}M^{\prime
2}t^{\prime}_{B}\Big{)}.$ (23)

Since the RHS of Eq. (22) depends only on $r$, while the LHS is a function of
$t,r$, the following solutions arise:

  1. (a)

$n=0$ and $A=0$,

  2. (b)

$m+n=1$ and $A=0$,

with case 1 being the unique TEGR solution, leaving case 2 as a non-trivial
and non-static exact inhomogeneous TG solution. This is an exact solution
since Eqs. (16)-(18) are immediately satisfied for $\dot{Y}\neq 0$ and with no
additional restrictions on the constants $m$ and $n$.

Conclusions.- We have derived and presented in this paper the first
inhomogeneous non-static exact solution in $f(T,B)$ gravity for the LTB
metric. We remark that (i) this is an exact solution for a form of $f(T,B)$
that is compatible with a viable cosmological scenario and (ii) the solution
corresponds to a spherically symmetric dust source that satisfies the same
evolution equation for the area radius Eq. (4) as its equivalent LTB models in
GR.

While we are now presenting a straightforward solution, we believe that it is
the first step that should serve to motivate the search for inhomogeneous
models in extended gravity theories that should be potentially useful for
cosmological and astrophysical applications. Given the utility of LTB models
in GR, the next step is to consider the general class of models (with $k\neq
0$, $\Lambda>0$) under a TG approach. An attempt to consider avoidance of
specific ansatzes or even considering a modified LTB metric should also be
done. This task is being currently undertaken and its results will be reported
elsewhere.

###### Acknowledgements.

CE-R acknowledges the Royal Astronomical Society as FRAS 10147. CE-R and A.A
are supported by PAPIIT-UNAM Project IA100220 and would like to acknowledge
networking support by the COST Action CA18108. SN acknowledges financial
support from SEP–CONACYT postgraduate grants program and RAS acknowledges
support from PAPIIT-DGAPA RR107015. We thank Jackson Levi Said for useful and
enlightening discussions.

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societal-issues |  Applications for solving societal issues such as human trafficking cliamate change etc.
brainstorming |  The answer mentions that the application’s goal is to provide a new angle to solve problems using brainstorming
business-productivity |  The answer mentions that the goal of the application is to help businesses to cut down on resources or create more output and profit
personal-life-productivity |  The answer mentions that the goal of the application is to make them more productive in their life personally or to save a lot of time for themselves
societal-productivity |  The answer mentions that the task will be more efficient and the application will help make the society more productive
accessibility-marginalized-disabled |  The participant answer mentions that the goal of the application is to marginalized and disabled people will be able to get help
lower-barriers-resources |  The participant answer mentions that the goal of the application is to lower barrier to resources
new-code |  None of the above codes apply but the answer is still meaningful so a new code is needed
na |  The participant answer does not make sense in the context
Table 19. Codes and their definition for Q11 & Q14 (harms of developing). Code |  Definition
---|---
manipulate-people |  Harm that misleads people to make choices that do not benefit them and is deceptive or fraudulent
misinformation |  Harm that causes people to believe in false information or incorrect state of the world by intentionally providing misleading knowledge i.e. spreading misinformation
bias |  Infringement of social justice by spreading prejudice and bias
mental-harm |  Mentally harm or upsets people by hurtful outputs and spread of negative information
overreliance |  People becoming dependent on technology and overtrusting and overrelying on them leading to diminished abilities to complete the task
physical-harm |  Technology leads to physical harm such as injuries and deaths
war |  Technology is used for wars or leads to wars and physical harm at a societal or global level
economic-disturbance |  Technology causing wider economic harms and disturbances such as widespread job loss or depression
financial-disturbance |  Technology causing more individual or smaller scale financial loss or property damage
human-labor-replacement |  Technology causes job loss and replacement of human labor force causing unemployment
social-isolation |  Technology causes weakened interpersonal connection especiallh with family and friends leading to isolation
range |  Technology causes a range of harms from very small impact to serious and more wide-spread harms
aid-criminal |  Technology is used to aid criminal activity
distrust-ai |  Technology or complicated output leads to distrust or underuse of the AI application
distrust-institution |  Technology leads to distrust of institutions such as the healthcare system
data-security-privacy-risk |  Privacy is invaded or data is lost through the use of technology or data is used in a negative way to benefit other stakeholders rather than the user
plagirism |  Technology plagirizes the existing work or copyrighted work
damaging-creativity |  Technology damages creativity or leads to unoriginality
hinder-career |  Technology causes career damage
incorrect-ai-output |  AI output being unintentionally incorrect or erroneous leads to different harms to users such as misdiagnosis or incorrect advice
legal-issues |  AI causing legal issues such as law suits due to illegal outputs
impede-learning |  AI causes people to not learn or grow as much
social-division |  AI causes social division and leads societal spread of hate or distrust
extinction |  AI leads to extinction of some sort such as group of people human race other animals or culture
minority |  AI leads to harming minority or underrepresente groups
waste-resources-or-time |  Technology leads to wasting resources such as time in development and is useless
unqualified-accessibility |  Technology makes certain tasks too easy so that non-qualified people or bad actors have better accessibility to these tasks
terrorism |  Technology leads to aiding terrorists for example in making weapons or bombs
business-use |  Technology is used by businesses to maximize profit
non-war-military-use |  Technology is used by the military for non-war purposes
lower-quality |  Technology lowers the quality by making mistakes or creating homogenious outputs which are worse than human work
information-access |  Technology prevents access to information
general-harm |  Risks security through lack of safety checks or the application is rendered unsafe and can harm or hurt users in unspecified ways
hinder-science |  Hinders scientific breakthroughs
no-harm |  No harms caused
miscommunication |  The use of application leads to miscommunication
negative-health-wellbeing |  Technology causes negative health outcome
hinder-medical-care |  Technology causes hindrance to medical and healthcare advancement and application
environmental-harm |  Causes environmental harm or allows continued environmental harm such as climate change
hacking-risk |  AI could be hacked by bad actors to be used for malicious tasks
new-code |  None of the above codes apply but the answer is still meaningful so a new code is needed
na |  The participant answer does not make sense in the context
Table 20. Codes and their definition for Q8 (benefits of developing). Code |  Definition
---|---
scientific-research-innovation |  Advances science research innovation and discovery
improve-medical-care |  Improves medical care
specialized-resources-accessibility |  Allows specialized resources such as medical mental legal or financial resources and services more available
information-accessibility |  Allows information to be more accessible
resource-accessibility |  Provides resources more accessible for everyday or life tasks
personal-life-efficiency |  Allows people to be more productive with less time and effort to solve tasks faster and easier
reduce-mundane-work |  Reduces mundane work in everyday life
improve-mental-health |  AI application improves mental health
companionship |  Provides companionship
social-interaction |  Provides assistance in social interaction
better-communication |  Allows people to communicate more effectively especially with less misunderstanding
more-production |  Application allows more production of goods or services through automation or reducing overhead of human labor
personal-growth |  Allows opportunities for personal growth or learning
enhancing-creativity |  AI tool inspires more creativity
financial-gain |  AI application leads to financial gain
safety |  AI helps make the world safer by handling dangerous situations or improving policing
less-human-error |  AI helps reduce human errors or bias
information-quality |  AI assists in improving information quality by fact-checking or ensuring that the infromation is correct
improve-well-being-health |  AI improves well-being fitness and health
improve-societal-issues |  AI improves societal issue
save-life |  The tecnology can save lives
general-efficiency |  The technology offers general efficiency in speeding up the process or reducing needed resources
no-benefit |  The answer says there is no benefit to the technology
new-code |  None of the above codes apply but the answer is still meaningful so a new code is needed
na |  The participant answer does not make sense in the context
Table 21. Codes and their definition for Q17 (harms of not developing). Code |  Definition
---|---
less-innovation |  Not using AI will lead to less innovation leading to stagnation of society
delay-in-innovation |  Not using AI will delay the rate of growth or breakthroughs however it might be possible for humans to get there without AI but just slower
misinformation |  Lack of AI usage continues division or mislead people with incorrect information
waste-resources-or-time |  Not using AI makes the task less efficient
business-loss |  Businesses or companies losing profit
financial-disturbance |  Financial loss at a smaller scale such as personal finance
unemployment |  People losing jobs
physical-harm |  Not using AI leads to physical harms such as death and injury
impede-personal-growth |  People lose the opportunity to grow or achieve without the help of AI
human-error |  Without AI human errors can be harmful
mental-harm |  Not developing the application leads to worse mental health such as anxiety depression loneliness etc.
health-issues |  Lack of AI assistance causes people to be unhealthy
stress-overworked |  Causing people to be overworked or be stressed because of the lack of automation offered by AI
inefficiency |  People will have to find another way that is not dependent on AI to solve the issue would cause some inconvenience but not disruptive
environmental-harm-continues |  Global warming and other environmental issue continues
lose-transparency |  Lack of AI assistance to understand complex systems leads to less transaprency
hinder-communication |  Without the application there will continue to be misunderstandings and difficulties in communication
hinder-creative-work |  Without the application products will become less creative limited to human creativity
lose-tech-race |  Lack of development will lead to losing technical race between countries and cause political tension
hinder-medical-care |  Not devloping the application will hinder patients from getting better medical care or treatment
lose-information-knowledge |  Not developing the application will lead to loss of information and knowledge
lose-accessibility-solution-service |  Not developing the application will lead to losing one of the solutions to a problem or service
lose-assistance |  Not developing the application will result in less help and assistance for the task
economic-distrubance |  Leads to economic distrubance such as cost increases at a larger scale
no-harm |  There is no harm of not developing the application
new-code |  None of the above codes apply but the answer is still meaningful so a new code is needed
na |  The participant answer does not make sense in the context
Table 22. Codes and their definition for Q20 (benefits of not developing). Code |  Definition
---|---
less-dependent-on-tech |  Make people less dependent on technology and self reliant in that they will have the skills to complete the tasks themselves
less-improper-unethical-use |  Generally reduces misuse or ethical concerns of AI
relieve-plagarism |  Relieves plagiarism issues
more-privacy |  Preserves data privacy
job-security |  Preserves employment and job security
learning-skills-knowledge |  People would learn more without AI
human-interaction-dependence |  People will interact more with real people and not AI increasing social interaction and interpersonal relations learning to depend on each other and invest in each other
environmental |  The environmental harms will be reduced
creativity |  Without AI people will be more creative and outputs will be more unique
less-misinformation |  Using AI for generation or spreading of misinformation would be avoided and reduced
maintain-status-quo |  Without the disruption of AI the current world will continue as is i.e. social order will not be disrupted and will continue to develop at the current pace
financial-benefit |  Without AI people will continue to pay for services and their providers as before increasing their financial benefit
empathy |  Without AI interactions and services will be more empathetic
higher-quality |  Humans will create higher quality outputs with less ai errors
better-health |  Results in better physical and mental health
human-control |  Humans will be able to control from their understanding of the process for certain tasks
more-attentive |  People will be more attentive to the task and lead to more understanding of the underlying problem
human-brilliance |  The world will rely more on human brilliance leading to more investment and celerbration of human ingenuity
other-non-ai-solutions |  Development will happen even if the application is banned through other non-ai solutions
no-benefit |  Participant answer specifies that there is no benefit
new-code |  None of the above codes apply but the answer is still meaningful so a new code is needed
na |  The participant answer does not make sense in the context
Table 23. Codes and their definition for coding groups (Q9, Q12, Q15, Q18, Q21). Code |  Definition
---|---
researchers-scholars |  People who are researchers (e.g. scientists) or scholars experts
low-skills |  People with limited skills education knowledge or critical-thinking
adults |  People who are adults but not elderly
youth |  Young people
seniors |  People who are old
lawyers |  People who are lawyers
lawyer-clients |  People who are clients of lawyers
rich |  People who are rich or in a high socioeconomic status
poor |  People who are not rich or engage in risky financial habits
middle-class |  People who are in the middle class
tech-access |  People who do have access to cutting-edge technology
no-tech-access |  People who do not have access to cutting-edge technology
it-professionals |  People who work in the IT industry such as software engineers
engineers |  People who are engineers
internet-users |  People who use the internet
anti-technology |  People who are skeptical of technology
teachers |  People who teach others
students |  People who are students or engage in learning
coaches |  People who are coaches
athletes |  People who are athletes
english-speakers |  People who only speak English for a language
nonenglish-speakers |  People who do not speak English
citizens |  People who are American citizens and have privileges only granted to these citizens such as voting
immigrants |  People who are immigrants
developing-nations |  People who live in developing nations
travelers |  People who travel or are interested in other cultures
democrats |  People who are Democrats or left-leaning
republicans |  People who are Republicans
doctors-nurses |  Healthcare professionals such as doctors and nurses
patients |  People who are patients or are receiving healthcare
businesses |  Businesses people who own businesses or high-level executives
consultants |  People who are consultants
consumers-stakeholders |  People who are consumers or stakeholders of a service or product
workers |  People who work professionally
busy-people |  People who are busy or have limited time due to other pressing commitments
disability-health-conditions |  People who have physical and mental disbilities or preexisting or chronic health conditions
nd-people |  People who are neurodivergent
mental-health |  People who have mental health conditions
single |  People who are not married or single
families |  A family
religious-minority |  People who belong to a minority religious gruop
christian |  People who are Christian
racial-minority |  People who are racial minorities and are not White
white |  People who are White
do-drive |  People who drive
don’t-drive |  People who do not drive or are passengers
men |  People who are men
gender-minority |  People who belong to a minority gender group
lgbtq |  People who belong to the LGBTQ community
government-officials |  Government institutions or people who work for such institutions
crime-victims |  People who are negatively impacted by a crime including crime victims victims’ families or people falsely accused of crimes
activists |  People who advocate for any social cause
criminals |  People who are criminals or engage in illegal activity
all |  All people in the world
minority-vulnerable |  People who belong to groups that are minoritized or generally vulnerable
na |  There is no valid group
remote-location |  People who live in remote areas
urban |  People who live in urban environments
artists-creatives |  People who are artists or engage in creative work
chefs |  People who cook
high-power |  People who have large amounts of influence or power
me |  The response references the person writing the response


received the Bachelor of Applied Science degree in Applied Math from Queen's University, Canada in 2017. He received the MSc degree in computer science from Ryerson University, Canada in 2020. He is currently a PhD student in the department of Electrical Engineering and Computer Science in York University, Toronto, Canada. He holds the position of Lead Scientist in Residence at NextAI and is a Postgraduate Affiliate at the Vector Institute. His main research field of interest is in the intersection of interpretable computer vision and video understanding.

[]Mennatullah Siam
received the B. Sc. degree from Computer Science in Ainshams University, Cairo in 2010, the MSc degree in Informatics from Nile University, Egypt in 2013 and the PhD degree from Computing Science department in University of Alberta, Edmonton, Canada in 2021. She is currently a postdoctoral researcher in the department of Electrical Engineering and Computer Science in York University, Toronto, Canada and a Vector institute affiliate. She was a recipient of Alberta Innovates Foundations Technology and Verna Tate graduate scholarships, and is currently a recipient of VISTA postdoc fellowship. Her major fields of interest are few-shot segmentation, video object segmentation, few-shot video understanding and spatiotemporal models interpretability.

[]Md Amirul Islam is a Senior Researcher at Noah's Ark Lab, Huawei Technologies Canada. He received his PhD in computer science from Ryerson University, Canada in 2022. He was a Postgraduate Affiliate at the Vector Institute. He received his M.Sc. in Computer Science from University of Manitoba, Canada in 2017 and his B.Sc. in Computer Science and Engineering from
North South University, Bangladesh in 2014. His research interests are in human-centric AI and multi-modal computer vision.

[]Neil D. B. Bruce graduated from the University of Guelph with a B.Sc. Double major in CS and Pure Mathematics. Dr. Bruce then attended the University of Waterloo for an M.A.Sc. in System Design Engineering and York University for a Ph.D. in Computer Science. Prior to joining Guelph he worked in the Department of Computer Science at Ryerson University. Prior to this Dr. Bruce worked at the University of Manitoba as Assistant then Associate Professor. Dr. Bruce has postdoctoral experience working at INRIA (France) and Epson Canada. He is the recipient of the Falconer Rh Young Researcher Award and is a Faculty Affiliate at the Vector Institute, Toronto. His research explores solutions to issues in computer vision, deep-learning, human perception, neuroscience and visual computing.

[]Richard P. Wildes
Richard P. Wildes (Member, IEEE) received the PhD degree from the Massachusetts Institute of Technology in 1989. Subsequently, he joined Sarnoff Corporation in Princeton, New Jersey, as a Member of the Technical Staff in the Vision Technologies Lab. In 2001, he joined the Department
of Electrical Engineering and Computer
Science at York University, Toronto, where he is
a Professor, a member of the Centre for Vision
Research and a Tier I York Research Chair. He also is a visiting research scientist at Samsung Artificial
Intelligence Center (SAIC), Toronto.
Honours include receiving a Sarnoff Corporation
Technical Achievement Award, the IEEE D.G. Fink Prize Paper Award
and twice giving invited presentations to the US National Academy of
Sciences. His main areas of research interest are computational vision,
especially video understanding, and artificial intelligence.

[]Konstantinos G. Derpanis
Kosta Derpanis received the Honours Bachelor of Science (BSc) degree in computer science with a Minor in mathematics from the University of Toronto, Canada, in 2000.  He received the MSc (supervisors Prof. John Tsotsos and Prof. Richard Wildes) and PhD (supervisor Prof. Richard Wildes) degrees in computer science from York University, Canada, in 2003 and 2010, respectively.  For his dissertation work, he received the Canadian Image Processing and Pattern Recognition Society (CIPPRS) Doctoral Dissertation Award 2010 Honourable Mention.  Subsequently, he was a postdoctoral researcher in the GRASP Laboratory at the University of Pennsylvania under the supervision of Prof. Kostas Daniilidis.  In 2012, he joined the Department of Computer Science at Ryerson University, Toronto, and later was promoted to Associate Professor with early tenure. In 2021, he joined the Department of Electrical Engineering and Computer Science at York University as an Associate Professor.  He also serves as a visiting research scientist at Samsung Artificial Intelligence Center (SAIC), Toronto. His main research field of interest is computer vision with emphasis on motion analysis and human motion understanding, and related aspects in image processing and machine learning.