# Rhombic staircase tableaux and Koornwinder polynomials

Sylvie Corteel  CNRS, IRIF, Université Paris Cité, Paris, France and
Department of Mathematics, University of California Berkeley, USA.
<EMAIL_ADDRESS>,  Olya Mandelshtam  Department of Combinatorics and
Optimization, University of Waterloo, Ontario, Canada<EMAIL_ADDRESS>and  Lauren Williams  Department of Mathematics, Harvard University,
Cambridge, MA<EMAIL_ADDRESS>
###### Abstract.

In this article we give a combinatorial formula for a certain class of
Koornwinder polynomials, also known as Macdonald polynomials of type
$\tilde{C}$. In particular, we give a combinatorial formula for the
Koornwinder polynomials
$K_{\lambda}=K_{\lambda}(z_{1},\dots,z_{N};a,b,c,d;q,t)$, where
$\lambda=(1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all
“open boundary ASEP polynomials” $F_{\mu}$, where $\mu$ is a composition in
$\\{-1,0,1\\}^{N}$; these polynomials are related to the nonsymmetric
Koornwinder polynomials $E_{\mu}$ up to a triangular change of basis. Our
formulas are in terms of _rhombic staircase tableaux_ , certain tableaux that
we introduced in previous work to give a formula for the stationary
distribution of the two-species asymmetric simple exclusion process (ASEP) on
a line with open boundaries.

###### Contents

  1. 1 Introduction
  2. 2 The Hecke algebra and Koornwinder polynomials
  3. 3 The proof of Theorem 1.10
  4. 4 Conclusion

##  1\. Introduction

In recent years there has been a great deal of activity connecting the
asymmetric simple exclusion process (ASEP), special functions, and
combinatorics, resulting in new combinatorial formulas for various special
functions. For example, Cantini, de Gier, and Wheeler [CdGW15] showed that the
specialization of the Macdonald polynomial $P_{\lambda}(x_{1}\dots,x_{N};q,t)$
at $x_{1}=\dots=x_{N}=1$ and $q=1$ is the partition function for the
_multispecies ASEP on a ring_. Subsequently Martin [Mar20] gave a formula for
the stationary distribution of the multispecies ASEP on a ring using some
combinatorial objects he called _multiline queues_. This led to our discovery
of a new formula for Macdonald polynomials and _ASEP polynomials_ , which are
related to the nonsymmetric Macdonald polynomials by a triangular change of
basis, in terms of multiline queues [CMW22].

A few years later, Ayyer, Martin, and Mandelshtam [AMM20, AMM22] proved
analogous results, connecting the _modified Macdonald polynomials_ , _queue-
inversion tableaux_ , and the TAZRP (totally asymmetric zero range process).
In particular, they gave a new formula for modified Macdonald polynomials in
terms of queue-inversion tableaux.

1234567$\ast$$\ast$${\alpha}$${\delta}$${t}$${t}$${1}$${1}$${t}$${\beta}$${\gamma}$

$\alpha$$\gamma$$\gamma$$\beta$$\delta$$t$$\alpha
t$$t^{2}$$\delta$$\gamma$$\beta t$$t$$t^{2}$$t$$t$$t$$t$$t$$t$

Figure 1. On the left: the two-species open ASEP state
$\mu=(\mathord{\bullet},\mathord{\ast},\mathord{\circ},\mathord{\ast},\mathord{\bullet},\mathord{\bullet},\mathord{\circ})$.
On the right: a rhombic staircase tableau of type $\mu$.

In this article we describe a parallel story, this time relating certain
Koornwinder polynomials $K_{\lambda}=K_{\lambda}(\mathbf{z};a,b,c,d;q,t)$,
rhombic staircase tableaux, and the open boundary ASEP. In [CGdGW16], Cantini,
Garbali, de Gier and Wheeler showed that the specialization of the Koornwinder
polynomial $K_{\lambda}$ at $z_{1}=\dots=z_{N}=1$ and $q=1$ is the partition
function for the _multispecies open ASEP_ (the multispecies ASEP on a line
with open boundaries), generalizing a previous result of Cantini [Can17] for
the two-species open ASEP. In [CMW17], we gave a formula for the stationary
distribution of the two-species open ASEP in terms of some tableaux we called
_rhombic staircase tableaux_. Building on the above insights, in this article
we give a combinatorial formula for the Koornwinder polynomials
$K_{\lambda}=K_{\lambda}(\mathbf{z};a,b,c,d;q,t)$, where
$\lambda=(1,\dots,1,0,\dots,0)$, by incorporating “spectral parameters”
$z_{1}^{\pm 1},\dots,z_{N}^{\pm 1}$ into the rhombic staircase tableaux. We
also give combinatorial formulas for the “open boundary ASEP polynomials”
$f_{\mu}(\mathbf{z};a,b,c,d;q,t)$ associated to compositions in
$\\{-1,0,1\\}^{N}$, which are analogously related to the nonsymmetric
Koornwinder polynomials $E_{\mu}(\mathbf{z};a,b,c,d;q,t)$ by a triangular
change of basis. While our results only treat a special case of Koornwinder
polynomials, this special case is already quite nontrivial; as we explain
below, Koornwinder polynomials represent a substantial generalization of
Macdonald polynomials.

In the remainder of the introduction we provide some background on Koornwinder
polynomials, the ASEP, and rhombic staircase tableaux, then state our main
result.

###  1.1. Koornwinder polynomials.

In [Mac88], Macdonald introduced a family of orthogonal symmetric polynomials
in the variables $\mathbf{x}=x_{1},x_{2},\ldots,x_{N}$ and parameters $q,t$
indexed by partitions $\lambda$, associated to different Lie types. In type
$A$, these are the classical Macdonald polynomials
$P_{\lambda}=P_{\lambda}(\mathbf{x};q,t)$, which have been studied extensively
in representation theory, algebraic combinatorics, and mathematical physics.
In type $\tilde{C}$, the Macdonald polynomials recover

the _Koornwinder polynomials_
$K_{\lambda}=K_{\lambda}(\mathbf{z};a,b,c,d;q,t)$ (sometimes called
_Macdonald-Koornwinder polynomials_), a 6-parameter family of symmetric
Laurent polynomials introduced by Koornwinder in [Koo92], and further studied
by van Diejen [vD95], Noumi [Nou95], Sahi [Sah99], and others. We note that
the Koornwinder polynomials are a multivariate generalization of the the well-
known Askey–Wilson polynomials, which in turn specialize or limit to all other
families of classical hypergeometric orthogonal polynomials in one variable
[AW85]. Moreover the Koornwinder polynomials give rise to the Macdonald
polynomials associated to any classical root system via a limit or
specialization; in particular, the usual (type A) Macdonald polynomial
$P_{\lambda}$ is the term of highest degree $|\lambda|$ in the Koornwinder
polynomial $K_{\lambda}$ [vD95].

To keep our notation compact, we will often write $K_{\lambda}$ or
$K_{\lambda}(\mathbf{z};q,t)$ for $K_{\lambda}(\mathbf{z};a,b,c,d;q,t)$. When
the parts of $\lambda$ are bounded by one, we will write
$K_{\lambda}(\mathbf{z};t)$, since $K_{\lambda}(\mathbf{z};q,t)$ is
independent of $q$.

There are also nonsymmetric versions of Macdonald and Koornwinder polynomials.
The _nonsymmetric Koornwinder polynomials_
$E_{\mu}=E_{\mu}(\mathbf{z};a,b,c,d;q,t)$, which are indexed by compositions
$\mu\in\mathbb{Z}^{N}$, were introduced by Sahi in [Sah99] as the joint
eigenfunctions of mutually commuting difference operators constructed from
generators of the affine Hecke algebra. For a partition $\lambda$, the
(symmetric) Koornwinder polynomial $K_{\lambda}$ can be obtained as a linear
combination of the $E_{\mu}$’s, where $\mu$ ranges over all signed
permutations of $\lambda$.

Although there has been a wealth of combinatorics developed to study
(symmetric and nonsymmetric) Macdonald polynomials in type $A$, such as the
tableaux formulas of Haglund, Haiman, and Loehr [HHL04], multiline queue
formulas [CMW22], and vertex model formulas [BW19], so far there has been much
less progress developing the combinatorics of Koornwinder polynomials. Ram and
Yip gave a type-independent formula for Macdonald polynomials in terms of
(quantum) alcove paths [RY11], which has been specialized to the Koornwinder
case by Orr and Shimozono, see [OS18, Theorem 3.13 and Proposition 3.20].
However, these formulas are much less explicit and combinatorial than their
type $A$ counterparts.

In this paper we aim to leverage the connection between Koornwinder
polynomials and the open boundary ASEP [Can17, CW18, CGdGW16, FV17] together
with the tableaux formula for the two-species open boundary ASEP [CMW17] to
give tableaux formulas for certain Koornwinder polynomials.

###  1.2. The two-species open boundary ASEP

The _asymmetric simple exclusion process (ASEP)_ is a canonical example of an
out-of-equilibrium system of interacting particles on a one-dimensional
lattice, originally introduced by Spitzer [Spi70]. In the ASEP with open
boundaries, the particles hop left and right on a one-dimensional finite
lattice, subject to the constraint that there is at most one particle at each
site; particles can also enter and exit the lattice at the left and right
endpoints of the lattice, at rates $\alpha,\beta,\gamma$ and $\delta$.

There is a multispecies version of the open boundary ASEP, in which particles
have _weights_ $\\{0,\pm 1,\ldots,\pm m\\}$. For the purpose of this article,
it is enough to define the two-species open boundary ASEP, shown at the left
of Figure 1.

######  Definition 1.1.

The _two-species asymmetric simple exclusion process (ASEP) with open
boundaries_ (or _two-species open ASEP_) is a model of interacting particles
on a one-dimensional lattice of $N$ sites, such that each site is either
vacant (represented by $\mathord{\circ}$ or “-1”) or occupied by a first class
particle (represented by $\mathord{\bullet}$ or “1”) or a second class
particle (represented by $\mathord{\ast}$ or “0”). States of the ASEP are
given by words
$\mu=(\mu_{1},\ldots,\mu_{N})\in\\{\mathord{\bullet},\mathord{\circ},\mathord{\ast}\\}^{N}=\\{-1,0,1\\}^{N}$.
We say that such a word $\mu$ has _length_ $|\mu|=N$. First class particles
may enter and exit at the left and right boundaries, and both types of
particles can hop left or right to adjacent (vacant) sites. The transitions in
this Markov chain are as follows:

  1. (1)

$X\mathord{\bullet}\mathord{\circ}Y\rightarrow
X\mathord{\circ}\mathord{\bullet}Y$,
$X\mathord{\bullet}\mathord{\ast}Y\rightarrow
X\mathord{\ast}\mathord{\bullet}Y$, and
$X\mathord{\ast}\mathord{\circ}Y\rightarrow X\mathord{\circ}\mathord{\ast}Y$
with probability $\frac{t}{N+1}$,

  2. (2)

$X\mathord{\circ}\mathord{\bullet}Y\rightarrow
X\mathord{\bullet}\mathord{\circ}Y$,
$X\mathord{\ast}\mathord{\bullet}Y\rightarrow
X\mathord{\bullet}\mathord{\ast}Y$, and
$X\mathord{\circ}\mathord{\ast}Y\rightarrow X\mathord{\ast}\mathord{\circ}Y$
with probability $\frac{1}{N+1}$,

  3. (3)

$\mathord{\circ}Z\rightarrow\mathord{\bullet}Z$ with probability
$\frac{\alpha}{N+1}$,

  4. (4)

$\mathord{\bullet}Z\rightarrow\mathord{\bullet}Z$ with probability
$\frac{\delta}{N+1}$,

  5. (5)

$Z\mathord{\bullet}\rightarrow Z\mathord{\circ}$ with probability
$\frac{\beta}{N+1}$,

  6. (6)

$Z\mathord{\circ}\rightarrow Z\mathord{\bullet}$ with probability
$\frac{\gamma}{N+1}$,

where $X,Y,Z\in\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{*}$
such that $|X|+|Y|=N-2$ and $|Z|=N-1$.

Note that when there is a $\mathord{\ast}$ at a boundary, no particles may
enter or exit at that boundary. In particular, the number of second class
particles is conserved.

Building upon earlier work of [Uch08], the first and third author discovered
that the partition function $Z_{N,r}$ of the two-species open ASEP on a
lattice of $N$ sites with $r$ second class particles can be described as a
_Koornwinder moment_ of type $\lambda=((N-r),0^{r})$ [CW18]. Concurrently,
Cantini [Can17] found that the partition function $Z_{N,r}$ coincides with the
specialization of a Koornwinder polynomial
$K_{\lambda}(\mathbf{z};a,b,c,d;q,t)$ of type $\lambda=(1^{N-r},0^{r})$ when
$z_{1}=\dots=z_{N}=1$ after a change of variables according to [Can17, (9)].
111We could also add here the condition $q=1$, but Koornwinder polynomials
$K_{(1,\dots,1,0,\dots,0)}$ in fact have no dependence on $q$. In subsequent
work, we introduced rhombic staircase tableaux in order to describe the
stationary distribution (and in particular the partition function) of the two-
species open ASEP [CMW17].

###  1.3. Rhombic staircase tableaux

In [CW11], Corteel and Williams introduced _staircase tableaux_ and used them
to give a combinatorial formula for the stationary distribution of the open
boundary ASEP.

Subsequently, Mandelshtam and Viennot [MV18] introduced _rhombic alternative
tableaux_ , and used them to give a combinatorial formula for the stationary
distribution of the two-species open ASEP, when $\gamma=\delta=0$. Finally in
[CMW17], we introduced _rhombic staircase tableaux_ , and used them to give a
combinatorial formula for the stationary distribution of the two-species open
ASEP (with all parameters general). We now define these tableaux.

######  Definition 1.2.

Let
$\mu=(\mu_{1},\ldots,\mu_{N})\in\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$.
We define the _rhombic diagram_ $\Gamma(\mu)$ of shape $\mu$ to be the
piecewise linear curve whose southeast border is obtained by reading $\mu$
from left to right and adding a south step followed by a west step for each
$\mathord{\circ}$ or $\mathord{\bullet}$, and a southwest step for each
$\mathord{\ast}$. We label the squares and rhombi adjacent to the southeast
border from top to bottom with $\\{1,\ldots,N\\}$. Then a square labelled $j$
corresponds to $\mu_{j}\in\\{\mathord{\circ},\mathord{\bullet}\\}$, and a
rhombus labeled $j$ corresponds to $\mu_{j}=\mathord{\ast}$. The northwest
border is obtained by adding $N-r$ west steps followed by $r$ southwest steps
followed by $N-r$ south steps to the northeast end of the southeast border,
where $r$ are the total numbers of $\mathord{\ast}$ particles in $\mu$. We say
such $\Gamma(\mu)$ has _size_ $(N,r)$.

See Figure 2 for an example.

######  Definition 1.3.

We can tile $\Gamma(\mu)$ using three types of tiles:

  * •

a _square_ tile consisting of horizontal and vertical edges,

  * •

a _horizontal rhombic_ tile (or “horizontal rhombus” or “short rhombus”)
consisting of horizontal and diagonal edges,

  * •

and a _vertical rhombic_ tile (or “vertical rhombus” or “tall rhombus”)
consisting of vertical and diagonal edges.

We will always choose a distinguished tiling of $\Gamma(\mu)$, shown in Figure
2.

1234567

Figure 2.  $\Gamma(\mu)$ and its distinguished tiling when
$\mu=(\mathord{\bullet},\mathord{\ast},\mathord{\circ},\mathord{\ast},\mathord{\bullet},\mathord{\bullet},\mathord{\circ})\in\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{7}$,
with labels assigned to the tiles on the southeast border.

######  Definition 1.4.

A _west-strip_ is the maximal contiguous strip consisting of squares and
vertical rhombic tiles, where all tiles are adjacent along a vertical edge.
Likewise, a _north-strip_ is the maximal contiguous strip consisting of
squares and horizontal rhombi, where all tiles are adjacent along a horizontal
edge. The total number of west- and north-strips in a rhombic diagram of size
$(N,r)$ is equal to $N-r$. The _distinguished tiling_ of a rhombic diagram is
the tiling in which all north strips consist of squares then horizontal rhombi
from bottom to top.

See Figure 2 for an example.

######  Definition 1.5.

For $\mu\in\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$, a
_rhombic staircase tableau (RST)_ of _type_ $\mu$ is a filling of the tiles of
$\Gamma(\mu)$ with the letters $\alpha,\beta,\gamma,\delta$ which satisfies
the following conditions:

  * •

A square on the southeast border corresponding to $\mu_{j}=\mathord{\circ}$
must contain a $\beta$ or a $\gamma$.

  * •

A square on the southeast border corresponding to $\mu_{j}=\mathord{\bullet}$
must contain an $\alpha$ or a $\delta$.

  * •

Each tile is either empty or contains one letter.

  * •

A horizontal rhombus may contain the letters $\alpha$ or $\gamma$.

  * •

A vertical rhombus may contain the letters $\beta$ or $\delta$.

  * •

Every tile in the same north-strip and above $\alpha$ or $\gamma$ must be
empty.

  * •

Every tile in the same west-strip and to the left of $\beta$ or $\delta$ must
be empty.

$\alpha$$\gamma$$\gamma$$\beta$$\delta$$\alpha$$\delta$$\gamma$$\beta$

$\alpha$$\gamma$$\gamma$$\beta$$\delta$$t$$\alpha
t$$t^{2}$$\delta$$\gamma$$\beta t$$t$$t^{2}$$t$$t$$t$$t$$t$$t$

Figure 3. On the left, we show a tableau of type
$\mu=(\mathord{\bullet},\mathord{\ast},\mathord{\circ},\mathord{\ast},\mathord{\bullet},\mathord{\bullet},\mathord{\circ})$,
and on the right we show the weights in $t$ associated to each tile. The total
weight of the tableau is $\alpha^{2}\beta^{2}\delta^{2}\gamma^{3}t^{14}$.

######  Definition 1.6.

The _weight_ $wt(T)$ of a rhombic staircase tableau $T$ is a monomial in
$\alpha,\beta,\gamma,\delta,t$, obtained by scanning each tile and giving it a
weight based on the nearest nonempty tile to its right (_resp._ below) in its
west-strip (_resp._ north-strip), if those exist:

  * •

Each horizontal rhombus containing $\alpha$ gets the weight $t$,

  * •

Each vertical rhombus containing $\beta$ gets the weight $t$,

  * •

Each empty square that sees $\alpha$ or $\gamma$ to its right and $\alpha$ or
$\delta$ below gets the weight $t$

  * •

Each empty square that sees $\beta$ to its right gets the weight $t$

  * •

Each empty vertical rhombus that sees $\beta$ to its right gets the weight
$t^{2}$

  * •

Each empty vertical rhombus that sees $\alpha$ or $\gamma$ to its right gets
the weight $t$

  * •

Each empty horizontal rhombus that sees $\alpha$ below gets the weight $t^{2}$

  * •

Each empty horizontal rhombus that sees $\beta$ or $\delta$ below gets the
weight $t$

Finally, $wt(T)$ is the product of all Greek letters in the filling of $T$
times the product of all weights in $t$ assigned to the tiles according to the
rules above. See Figure 3 for an example of a tableau of weight
$\alpha^{2}\beta^{2}\delta^{2}\gamma^{3}t^{14}$.

######  Definition 1.7.

Given a word
$\mu=(\mu_{1},\dots,\mu_{N})\in\\{-1,0,1\\}^{N}=\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$,
let $R(\mu)=\sum_{T}wt(T)$, where the sum is over all rhombic staircase
tableaux of type $\mu$. In particular, $R(\mu)$ is a polynomial in
$\alpha,\beta,\gamma,\delta$ and $t$.

######  Example 1.8.

The following figure shows all rhombic staircase tableaux of type
$\mathord{\bullet}\mathord{\bullet}$:

$\alpha$$\delta$$\alpha$$\delta$$\alpha$$\delta$$\alpha$$\delta$$\alpha$$\delta$$\alpha$$\delta$$\delta$$\delta$$\alpha$$\alpha$$\alpha$$\beta$$\gamma$$\delta$$t$$t$

Thus
$R(\mathord{\bullet}\mathord{\bullet})=\alpha\delta(1+t+\alpha+\beta+\gamma+\delta)+\alpha^{2}t+\delta^{2}.$

To give a second example, the following figure shows all rhombic staircase
tableaux of type $\mathord{\circ}\mathord{\ast}$:

$\beta$$\gamma$$\gamma$$\gamma$$t^{2}$$t$$\beta t$$\delta$

Thus $R(\mathord{\circ}\mathord{\ast})=\beta t^{2}+\gamma t+\beta\gamma
t+\gamma\delta$.

The main result of [CMW17] was the following.

######  Theorem 1.9.

[CMW17] Consider the two-species open boundary ASEP on a lattice of $N$ sites,
with $r$ second class particles. Let $Z_{N,r}=\sum_{\sigma}R(\sigma),$ where
the sum is over all words $\sigma\in\\{-1,0,1\\}^{N}$ containing exactly $r$
$0$’s. Then $Z_{N,r}$ is the partition function for the two-species open
boundary ASEP, and for each
$\mu\in\\{-1,0,1\\}^{N}=\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$
containing exactly $r$ $0$’s (or $\mathord{\ast}$’s), the steady state
probability of being in state $\mu$ equals

$\frac{R(\mu)}{Z_{N,r}}.$

Using Theorem 1.9, it now follows from the result

of Cantini [Can17] that $Z_{N,r}$ is the specialization of a Koornwinder
polynomial at $z_{1}=\dots=z_{N}$ and $q=1$. This gives rise to the natural
question: how can we incorporate the “spectral parameters” $z_{1},\dots,z_{N}$
into the rhombic staircase tableaux, so as to give a formula for the
Koornwinder polynomials themselves?

###  1.4. Main result

The above question leads us to our main result, described in Theorem 1.10,
which is a formula for

each “open boundary ASEP polynomial” $F_{\mu}(\mathbf{z};t)$, where $\mu$ is a
composition in $\\{-1,0,1\\}^{N}$. We note that when $\mu=((-1)^{N-r},0^{r})$,
$F_{\mu}(\mathbf{z};t)$ coincides with the nonsymmetric Koornwinder polynomial
$E_{\mu}(\mathbf{z};t)$. Additionally the sum over all
$\mu\in\\{-1,0,1\\}^{N}$ with $r$ 0’s is the symmetric Koornwinder polynomial
$K_{1^{N-r},0^{r}}(\mathbf{z};t)$, so we get combinatorial formulas for these
polynomials as well.

Recall from Definition 1.7 that

for
$\mu=(\mu_{1},\dots,\mu_{N})\in\\{-1,0,1\\}^{N}=\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$,
$R(\mu)$ denotes the generating polynomial in the variables
$\alpha,\beta,\gamma,\delta$ and $t$ for the rhombic staircase tableaux of
type $\mu$. We also let

(1.1) $\tilde{R}(\mu)=\frac{(t-1)^{N-r}}{\prod_{i=2r}^{N+r-1}(\alpha\beta
t^{i}-\gamma\delta)}R(\mu),$

where $r$ is the number of zeroes (or $\mathord{\ast}$) in $\mu$.

Let $\mu$ be a word of length $N:=|\mu|$, and let $[N]=\\{1,\ldots,N\\}$. For
subsets of indices $I,T\subseteq[N]$ and $u=\mu|_{I}$ a subword of $\mu$
restricted to the indices $I$, define $u|_{T}:=\mu|_{T\cap I}$ to be the
maximal subword of $u$ supported on $T$. For example, if
$\mu=\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}$, $I=\\{2,4,5\\}$,
$u=\mu_{2}\mu_{4}\mu_{5}$, and $T=\\{1,2,4\\}$, then $u|_{T}=\mu_{2}\mu_{4}$.

We introduce the following change of variables using the parameters
$a,b,c,d$222This change of variables is nearly the same as [Can17, (9)] but is
different from [CGdGW16, (24), (25)]).

$\displaystyle\alpha$ $\displaystyle=\frac{-ac(1-t)}{(a-1)(c-1)},$
$\displaystyle\qquad\gamma$ $\displaystyle=\frac{1-t}{(a-1)(c-1)}$ (1.2)
$\displaystyle\beta$ $\displaystyle=\frac{-bd(1-t)}{(b-1)(d-1)},$
$\displaystyle\qquad\delta$ $\displaystyle=\frac{1-t}{(b-1)(d-1)}$

Let $W_{0}=\langle s_{1},\dots,s_{N}\rangle$ be the finite Weyl group of type
$C_{N}$, as in Definition 2.1. For $S\subseteq[N]$, we let
$\overline{S}=[N]\setminus S$.

######  Theorem 1.10.

Let $\lambda\in\\{1,0\\}^{N}$ be a partition, and let $\boldsymbol{\delta}$ be
the signed permutation of $\lambda$ such that
$\boldsymbol{\delta}_{1}\leq\boldsymbol{\delta}_{2}\leq\dots\leq\boldsymbol{\delta}_{N}\leq
0$. Choose $\mu=(\mu_{1},\dots,\mu_{N})\in
W_{0}\cdot\lambda\subset\\{-1,0,1\\}^{N}=\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}^{N}$
and let $V=\\{i\ |\ \mu_{i}\in\\{\pm 1\\}\\}$.

We define

(1.3) $F_{\mu}(\mathbf{z};t):=\sum_{S\subseteq
V}\tilde{R}(\mu|_{\overline{S}})\cdot\prod_{i\in
S}(z_{i}^{\mu_{i}}-1)=\sum_{S\subseteq[N]}\tilde{R}(\mu|_{\overline{S}})\cdot\prod_{i\in
S}(z_{i}^{\mu_{i}}-1).$

Then as $\mu$ ranges over $W_{0}\cdot\lambda$, the Laurent polynomials

$\\{F_{\mu}(\mathbf{z};t)\ |\ \mu\in W_{0}\cdot\lambda\\}$

form a _qKZ family_ (in the sense of Definition 2.4).

Moreover, if we use the change of variables from (1.2),

  * •

$F_{\boldsymbol{\delta}}(\mathbf{z};t)$

equals the nonsymmetric Koornwinder polynomial
$E_{\boldsymbol{\delta}}(\mathbf{z};a,b,c,d;t)$, and

  * •

the symmetric Koornwinder polynomial $K_{\lambda}$ is equal to

$K_{\lambda}(\mathbf{z};a,b,c,d;q,t)=\sum_{\mu\in
W_{0}\cdot\lambda}F_{\mu}(\mathbf{z};t),$

where the sum runs over all distinct signed permutations of $\lambda$.

######  Remark 1.11.

We refer to the Laurent polynomials $\\{F_{\mu}(\mathbf{z};t)\\}$ as _open
boundary ASEP polynomials_ ; each $F_{\mu}$ specializes at
$z_{1}=\dots=z_{N}=1$ to the steady state probability that the two-species
ASEP is in state $\mu$.

######  Remark 1.12.

It follows from the definition of $F_{\mu}$ that the coefficient of $z^{\mu}$
in $F_{\mu}$ is $1$: we get this term when $S=V$ because $\tilde{R}(\mu)=1$
when $\mu=\mathord{\ast}^{r}$.

######  Remark 1.13.

We refer to the Laurent polynomials $F_{\mu}(z_{1},\dots,z_{N};t)$ as _open
boundary ASEP polynomials_ because they are the open boundary analogue of the
_ASEP polynomials_ (on a ring) which were first studied in [CdGW15, CMW22]
(and which were subsequently dubbed “ASEP polynomials” in [CdGW18]). We note
that the Laurent polynomials $F_{\mu}$ are related to the nonsymmetric
Koornwinder polynomials $E_{\mu}$ via a triangular change of basis, see
Proposition 2.9.

######  Example 1.14.

For $N=2$, the ASEP polynomials are:

  * •

$F_{(1,1)}=\widetilde{R}(\mathord{\bullet}\mathord{\bullet})+\widetilde{R}(\mathord{\bullet})(z_{1}+z_{2}-2)+(z_{1}-1)(z_{2}-1)$.

  * •

$F_{(1,-1)}=\widetilde{R}(\mathord{\bullet}\mathord{\circ})+\widetilde{R}(\mathord{\bullet})(z_{2}^{-1}-1)+\widetilde{R}(\mathord{\circ})(z_{1}-1)+(z_{1}-1)(z_{2}^{-1}-1)$.

  * •

$F_{(-1,1)}=\widetilde{R}(\mathord{\circ}\mathord{\bullet})+\widetilde{R}(\mathord{\circ})(z_{2}-1)+\widetilde{R}(\mathord{\bullet})(z_{1}^{-1}-1)+(z_{1}^{-1}-1)(z_{2}-1)$.

  * •

$F_{(-1,-1)}=\widetilde{R}(\mathord{\circ}\mathord{\circ})+\widetilde{R}(\mathord{\circ})(z_{1}^{-1}+z_{2}^{-1}-2)+(z_{1}^{-1}-1)(z_{2}^{-1}-1)$.

  * •

$F_{(1,0)}=\widetilde{R}(\mathord{\bullet}\mathord{\ast})+\widetilde{R}(\mathord{\ast})(z_{1}-1)$.

  * •

$F_{(0,1)}=\widetilde{R}(\mathord{\ast}\mathord{\bullet})+\widetilde{R}(\mathord{\ast})(z_{2}-1)$.

  * •

$F_{(-1,0)}=\widetilde{R}(\mathord{\circ}\mathord{\ast})+\widetilde{R}(\mathord{\ast})(z_{1}^{-1}-1)$.

  * •

$F_{(0,-1)}=\widetilde{R}(\mathord{\ast}\mathord{\circ})+\widetilde{R}(\mathord{\ast})(z_{2}^{-1}-1)$.

  * •

$F_{(0,0)}=\widetilde{R}(\mathord{\ast}\mathord{\ast})$.

For example, since
$\widetilde{R}(\mathord{\bullet}\mathord{\bullet})=\frac{(t-1)^{2}}{(\alpha\beta
t-\gamma\delta)(\alpha\beta-\gamma\delta)}\left(\alpha\delta(1+t+\alpha+\beta+\gamma+\delta)+\alpha^{2}t+\delta^{2}\right)$
and
$\widetilde{R}(\mathord{\bullet})=\frac{t-1}{\alpha\beta-\gamma\delta}(\alpha+\delta)$,
we have that

$F_{(1,1)}=\frac{(t-1)^{2}}{(\alpha\beta
t-\gamma\delta)(\alpha\beta-\gamma\delta)}\left(\alpha\delta(1+t+\alpha+\beta+\gamma+\delta)+\alpha^{2}t+\delta^{2}\right)\\\
+\frac{t-1}{\alpha\beta-\gamma\delta}(\alpha+\delta)(z_{1}+z_{2}-2)+(z_{1}-1)(z_{2}-1).$

We then have that $K_{(1,1)}=F_{(1,1)}+F_{(1,-1)}+F_{(-1,1)}+F_{(-1,-1)}$, and
$K_{(1,0)}=F_{(1,0)}+F_{(-1,0)}+F_{(0,1)}+F_{(0,-1)}$.

Using Theorem 1.10, we can also give a simpler combinatorial formula for
$K_{\lambda}(\mathbf{z};q,t)$, where $\lambda=(1^{N-r},0^{r})$.

######  Theorem 1.15.

Let $\lambda=(1^{N-r},0^{r})$ and let

$\tilde{Z}_{N,r}=\frac{(t-1)^{N-r}}{\prod_{i=2r}^{N+r-1}(\alpha\beta
t^{i}-\gamma\delta)}Z_{N,r}(\alpha,\beta,\gamma,\delta;t),$

where $Z_{N,r}=Z_{N,r}(\alpha,\beta,\gamma,\delta;t)$ is defined in Theorem
1.9 as the generating polynomial for staircase tableaux of size $N$ with $r$
diagonal steps. Then

(1.4) $K_{\lambda}(\mathbf{z};q,t)=\sum_{k=0}^{N-r}\tilde{Z}_{N-k,r}\cdot
e_{k}(y_{1},\ldots,y_{N}),$

where $y_{i}=z_{i}+1/z_{i}-2$ for $1\leq i\leq N$ and
$e_{k}(y_{1},\ldots,y_{N})$ is the classical elementary symmetric polynomial.

Now recall the fact that the Koornwinder polynomials
$K_{\lambda}(\mathbf{z};q,t)$ factor when $q=1$.

######  Proposition 1.16 ([RW16]).

Let $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ be any partition. Then at
$q=1$, we have the following formula for the Koornwinder polynomial
$K_{\lambda}$:

$K_{\lambda}(\mathbf{z};1,t)=\prod_{i=1}^{\lambda_{1}}K_{(1^{\lambda_{i}^{\prime}},0^{N-\lambda^{\prime}_{i}})}(\mathbf{z};t)=\prod_{i=1}^{\lambda_{1}}K_{1^{\lambda_{i}^{\prime}}}(\mathbf{z};t),$

where $\lambda^{\prime}$ is the partition conjugate to $\lambda$.

This allows us to give the following combinatorial formula for
$K_{\lambda}(\mathbf{z};1,t)$ for any partition $\lambda$.

######  Corollary 1.17.

The Koornwinder polynomials $K_{\lambda}(\mathbf{z};q,t)$ at $q=1$ can be
written as

$K_{\lambda}(\mathbf{z};1,t)=\sum_{\mu}e_{\mu}(y_{1},\ldots,y_{N})\prod_{i=1}^{\lambda_{1}}\tilde{Z}_{N-\mu_{i},N-\lambda^{\prime}_{i}},$

where $y_{i}=z_{i}+1/z_{i}-2$, and the sum is over compositions
$\mu=(\mu_{1},\ldots,\mu_{\ell})$ such that
$0\leq\mu_{i}\leq\lambda^{\prime}_{i}$ for $1\leq i\leq\lambda_{1}$.

We can consider Corollary 1.17 as giving a combinatorial formula for
Koornwinder polynomials in terms of sequences of $\ell$ rhombic staircase
tableaux.

The structure of this paper is as follows.

In Section 2 we define the affine Hecke algebra of type C, Koornwinder
polynomials, and the notion of a qKZ family. In Section 3 we prove our main
result by showing that our polynomials $\\{F_{\mu}\\}$ form a qKZ family. And
in Section 4 we compare this story to its counterpart in type A, and discuss
further directions and generalizations.

Acknowledgements:  We would like to thank the _Research in Paris_ program at
the Institut Henri Poincaré and the visiting scholars program at Université
Paris Cité which made this collaboration possible. We are grateful to Luigi
Cantini, Eric Rains, Siddharth Sahi, and Ole Warnaar for useful discussions.
SC was partially supported by NSF grant DMS-2054482, ANR grants
ANR-19-CE48-0011 and ANR-18-CE40-0033. OM was partially supported by NSERC
grant RGPIN-2021-02568. LW was partially supported by the National Science
Foundation under Award No. DMS-2152991. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the National Science
Foundation.

##  2\. The Hecke algebra and Koornwinder polynomials

In this section we review the affine Hecke algebra of type C, nonsymmetric and
symmetric Koornwinder polynomials, the notion of qKZ family, and we explain
the relationship between Koornwinder polynomials and qKZ family. For more
background on these topics, see [Lus89], [Koo92], [Sah99], and [NNSY01].

###  2.1. The affine Hecke algebra of type C

######  Definition 2.1.

The _affine Weyl group_ $\mathcal{W}_{N}$ of type $\tilde{C}_{N}$ is the group
generated by the elements $s_{0},s_{1},\dots,s_{N}$ subject to the relations

$\displaystyle s_{i}^{2}$ $\displaystyle=1\text{ for all }0\leq i\leq N$
$\displaystyle s_{i}s_{j}$ $\displaystyle=s_{j}s_{i}\text{ if }|i-j|>1$
$\displaystyle s_{i}s_{i+1}s_{i}$ $\displaystyle=s_{i+1}s_{i}s_{i+1}\text{ for
}1\leq i\leq N-2$ $\displaystyle s_{0}s_{1}s_{0}s_{1}$
$\displaystyle=s_{1}s_{0}s_{1}s_{0}$ $\displaystyle s_{N}s_{N-1}s_{N}s_{N-1}$
$\displaystyle=s_{N-1}s_{N}s_{N-1}s_{N}.$

The _finite Weyl group of type $C_{N}$_ is $W_{0}=\langle
s_{1},\dots,s_{N}\rangle$.

Given a parameter $q$, the affine Weyl group $\mathcal{W}_{N}$ acts on the
space $\mathbb{C}[z_{1}^{\pm 1},\ldots,z_{N}^{\pm 1}]$ of Laurent polynomials
in $N$ variables as follows:

$\displaystyle s_{i}f(z_{1},\ldots,z_{N})$
$\displaystyle=f(z_{1},\ldots,z_{i+1},z_{i},\ldots,z_{N}),\qquad 1\leq i\leq
N-1$ $\displaystyle s_{0}f(z_{1},\ldots,z_{N})$
$\displaystyle=f(qz_{1}^{-1},z_{2},\ldots,z_{N})$ $\displaystyle
s_{N}f(z_{1},\ldots,z_{N})$ $\displaystyle=f(z_{1},\ldots,z_{N-1},z_{N}^{-1})$

The affine Hecke algebra $\mathcal{H}_{N}$ of type $\tilde{C}_{N}$ is a
deformation of the group algebra of $\mathcal{W}_{N}$, which depends on three
parameters $t_{0},t_{N}$ and $t$. It is generated by elements
$T_{0},T_{1},\dots,T_{N}$ subject to the relations

(2.1) $\displaystyle T_{i}-T_{i}^{-1}$
$\displaystyle=t_{i}^{1/2}-t_{i}^{-1/2}\text{ for }0\leq i\leq N$ (2.2)
$\displaystyle T_{i}T_{j}$ $\displaystyle=T_{j}T_{i}\text{ if }|i-j|>1$ (2.3)
$\displaystyle T_{i}T_{i+1}T_{i}$ $\displaystyle=T_{i+1}T_{i}T_{i+1}\text{ for
}1\leq i\leq N-2$ (2.4) $\displaystyle T_{0}T_{1}T_{0}T_{1}$
$\displaystyle=T_{1}T_{0}T_{1}T_{0}$ (2.5) $\displaystyle
T_{N}T_{N-1}T_{N}T_{N-1}$ $\displaystyle=T_{N-1}T_{N}T_{N-1}T_{N}$

where $t_{1}=t_{2}=\dots=t_{N-1}=t$.

For $1\leq i\leq N$, define the operators

(2.6) $Y_{i}=(T_{i}\dots T_{N-1})(T_{N}\dots T_{0})(T_{1}^{-1}\dots
T_{i-1}^{-1}),$

which form an abelian subalgebra, and hence share a common set of
eigenfunctions, which are precisely the nonsymmetric Koornwinder polynomials
in the polynomial representation of the Hecke algebra $\mathcal{H}_{N}$.

The following operators, introduced by Noumi [Nou95], give a polynomial
representation of $\mathcal{H}_{N}$. These operators also appear as the
$T_{i}$’s in [CGdGW16, (73)] and are closely related to the $\hat{T}_{i}$’s in
[Can17] (up to swapping $b$ and $c$).

######  Definition 2.2.

We fix parameters $a,b,c,d,t,q$, as well as positive integer $N$. The
following operators act on the space $\mathbb{C}[z_{1}^{\pm
1},\dots,z_{N}^{\pm 1}]$ of Laurent polynomials in $N$ variables.

(2.7) $\displaystyle\widetilde{T}_{0}$
$\displaystyle=-\frac{ac}{q}-\frac{(z_{1}-a)(z_{1}-c)}{z_{1}}\cdot\frac{1-s_{0}}{z_{1}-qz_{1}^{-1}}$
(2.8) $\displaystyle\widetilde{T}_{i}$
$\displaystyle=t-(tz_{i}-z_{i+1})\cdot\frac{1-s_{i}}{z_{i}-z_{i+1}}$ (2.9)
$\displaystyle=s_{i}+\frac{(t-1)(z_{i}s_{i}-z_{i+1})}{z_{i}-z_{i+1}}\hskip
28.45274pt\text{ for }1\leq i<N$ (2.10) $\displaystyle\widetilde{T}_{N}$
$\displaystyle=-bd+\frac{(bz_{N}-1)(dz_{N}-1)}{z_{N}}\cdot\frac{1-s_{N}}{z_{N}-z_{N}^{-1}}.$

It is straightforward to check that these operators satisfy the relations
(2.1) to (2.5).

###  2.2. Koornwinder polynomials

We follow the exposition of [Kas11, Definition 4.1] and [CGdGW16, Definition
1] for the following characterization of nonsymmetric Koornwinder polynomials.

######  Definition 2.3.

[CGdGW16, Definition 1] Let $\lambda\in\mathbb{Z}^{N}$ be a composition, and
let $\lambda^{+}$ denote the unique dominant element in $W_{0}\cdot\lambda$,
that is, $\lambda_{1}^{+}\geq\lambda_{2}^{+}\geq\dots\geq\lambda_{N}^{+}\geq
0$. Take the shortest element $w\in W_{0}$ such that
$w\cdot\lambda^{+}=\lambda$, and denote it by $w^{+}_{\lambda}$. Let
$\rho=(N-1,N-2,\dots,1,0)$, and $\rho(\lambda)=w_{\lambda}^{+}\cdot\rho$.

Then the nonsymmetric Koornwinder polynomial $E_{\lambda}$ is the unique
polynomial which solves the eigenvalue equations

(2.11) $Y_{i}E_{\lambda}=y_{i}(\lambda)E_{\lambda}\text{ for }i=1\dots
N,\text{ where }$ (2.12) $t_{0}=-acq^{-1},\hskip 28.45274ptt_{N}=-bd,\text{
and }$ (2.13)
$y_{i}(\lambda)=q^{\lambda_{i}}t^{N-i+\rho(\lambda)_{i}}(t_{0}t_{N})^{\epsilon_{i}(\lambda)}\text{
and }\epsilon_{i}(\lambda)=\begin{cases}1\text{ if }\lambda_{i}\geq 0\\\
0\text{ if }\lambda_{i}<0\end{cases}$

and whose coefficient of the term $z^{\lambda}=z_{1}^{\lambda_{1}}\dots
z_{N}^{\lambda_{N}}$ is equal to $1$.

######  Definition 2.4.

[Kas11, Definition 3.1] Let $\lambda\in\mathbb{Z}^{N}$ be a composition.
Suppose that for each signed permutation $\mu$ of $\lambda$, we have a Laurent
polynomial $f_{\mu}\in\mathbb{C}[z_{1}^{\pm 1},\dots,z_{N}^{\pm 1}]$ such that
the following relations hold:

(2.14) $\displaystyle\widetilde{T}_{0}f_{\mu_{1},\dots}$
$\displaystyle=q^{\mu_{1}}f_{-\mu_{1},\dots}\text{ if }\mu_{1}<0$ (2.15)
$\displaystyle\widetilde{T}_{0}f_{\mu_{1},\dots}$
$\displaystyle=t_{0}f_{\mu_{1},\dots}\text{ if }\mu_{1}=0$ (2.16)
$\displaystyle\widetilde{T}_{i}f_{\dots,\mu_{i},\mu_{i+1},\dots}$
$\displaystyle=tf_{\dots,\mu_{i},\mu_{i+1},\dots}\text{ if }\mu_{i}=\mu_{i+1}$
(2.17) $\displaystyle\widetilde{T}_{i}f_{\dots,\mu_{i},\mu_{i+1},\dots}$
$\displaystyle=f_{\dots,\mu_{i+1},\mu_{i},\dots}\text{ if }\mu_{i}>\mu_{i+1}$
(2.18) $\displaystyle\widetilde{T}_{N}f_{\dots,\mu_{N}}$
$\displaystyle=t_{N}f_{\dots,\mu_{N}}\text{ if }\mu_{N}=0$ (2.19)
$\displaystyle\widetilde{T}_{N}f_{\dots,\mu_{N}}$
$\displaystyle=f_{\dots,-\mu_{N}}\text{ if }\mu_{N}>0$

Then we call the polynomials $\\{f_{\mu}\\}$ a _qKZ family_.

The following result appears in [CGdGW16], see [CGdGW16, Lemma 3] and the
discussion that follows. We take Proposition 2.5 as the definition of the
symmetric Koornwinder polynomial.

######  Proposition 2.5.

Let $\lambda=(\lambda_{1},\dots,\lambda_{N})$ be a partition. Suppose that we
have a qKZ family $\\{f_{\mu}\\}$ as in Definition 2.4, and suppose further
that for $\delta:=-\lambda$, we have that $f_{\delta}$ equals the nonsymmetric
Koornwinder polynomial $E_{\delta}$. Then the symmetric Koornwinder polynomial
$K_{\lambda}$ is equal to

$K_{\lambda}(z_{1},\dots,z_{N};q,t)=\sum_{\mu\in
W_{0}\cdot\lambda}f_{\mu}(z_{1},\dots,z_{n};q,t),$

where the sum runs over all distinct signed permutations of $\lambda$.

The following result will be a key ingredient in the proof of our main result.

######  Proposition 2.6.

Let $\lambda\in\mathbb{Z}^{N}$ be a composition, and let $\boldsymbol{\delta}$
be the signed permutation of $\lambda$ such that
$\boldsymbol{\delta}_{1}\leq\boldsymbol{\delta}_{2}\leq\dots\leq\boldsymbol{\delta}_{N}\leq
0$. Suppose that we have a qKZ family $\\{f_{\mu}\\}$ as in Definition 2.4.
Then

$Y_{i}f_{\boldsymbol{\delta}}=y_{i}(\boldsymbol{\delta})f_{\boldsymbol{\delta}}$

for $i=1,\dots,N$, i.e. (2.11) holds with $f_{\boldsymbol{\delta}}$ in place
of $E_{\lambda}$. Therefore if the coefficient of the term
$z^{\boldsymbol{\delta}}$ in $f_{\boldsymbol{\delta}}$ is equal to $1$, then
$f_{\boldsymbol{\delta}}$ equals the nonsymmetric Koornwinder polynomial
$E_{\boldsymbol{\delta}}$.

######  Example 2.7.

As an example, we compute $\tilde{R}(\mathord{\circ}\mathord{\ast})$. From
(1.3), we have that
$F_{\mathord{\circ}\mathord{\ast}}=(z_{1}^{-1}-1)\tilde{R}(\mathord{\ast})+\tilde{R}(\mathord{\circ}\mathord{\ast})=(z_{1}^{-1}-1)+\tilde{R}(\mathord{\circ}\mathord{\ast})$.
From Proposition 2.6,
$Y_{1}(F_{\mathord{\circ}\mathord{\ast}})=y_{i}F_{\mathord{\circ}\mathord{\ast}}=q^{-1}F_{\mathord{\circ}\mathord{\ast}}$,
where $Y_{1}=T_{1}T_{2}T_{1}T_{0}$ for $N=2$. We compute

$\displaystyle T_{1}T_{2}T_{1}T_{0}(z_{1}^{-1}-1)$
$\displaystyle=q^{-1}\left((z_{1}^{-1}-1)-\frac{(t-1)(\beta
t^{2}+\gamma\delta+\beta\gamma t+\gamma t)}{\gamma\delta}\right)$
$\displaystyle T_{1}T_{2}T_{1}T_{0}(\tilde{R}(\mathord{\circ}\mathord{\ast}))$
$\displaystyle=\frac{\alpha\beta t^{2}}{\gamma\delta
q}\tilde{R}(\mathord{\circ}\mathord{\ast})$

and thus

$\displaystyle T_{1}T_{2}T_{1}T_{0}(F_{\mathord{\circ}\mathord{\ast}})$
$\displaystyle=q^{-1}\left((z_{1}^{-1}-1)-\frac{(t-1)(\beta
t^{2}+\gamma\delta+\beta\gamma t+\gamma t)}{\gamma\delta}+\frac{\alpha\beta
t^{2}}{\gamma\delta}\tilde{R}(\mathord{\circ}\mathord{\ast})\right)$
$\displaystyle=q^{-1}\left((z_{1}^{-1}-1)+\tilde{R}(\mathord{\circ}\mathord{\ast})\right)$

Solving for $\tilde{R}(\mathord{\circ}\mathord{\ast})$, we get

$\tilde{R}(\mathord{\circ}\mathord{\ast})=\frac{(t-1)(\beta t^{2}+\gamma
t+\gamma\beta t+\delta\gamma)}{\alpha\beta t^{2}-\gamma\delta},$

which matches the combinatorial computation for
$R(\mathord{\circ}\mathord{\ast})$ from Example 1.8.

###### Proof.

We start by writing
$\boldsymbol{\delta}=(\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})$,
where $\lambda_{1}<\cdots<\lambda_{r}<0$ and $(k_{1},\ldots,k_{r})$ is a
(strong) composition of $k$, i.e. $\sum k_{i}=k$, and $k_{i}>0$ for all $i$.
So
$\boldsymbol{\delta}^{+}=(-\lambda_{1}^{k_{1}},\ldots,-\lambda_{r}^{k_{r}},0^{N-k})$.
The permutation $w_{\boldsymbol{\delta}}^{+}$ is the signed permutation which
divides $\\{1,2,\dots,N\\}$ into consecutive blocks of sizes
$k_{1},k_{2},\dots,k_{r},N-k$, and on the first $r$ blocks, it reverses then
negates the elements, and on the last block, it acts as the identify. (E.g. if
$\boldsymbol{\delta}=(-3,-3,-1,-1,-1,0,0)$, then $w_{\boldsymbol{\delta}}^{+}$
is the signed permutation mapping $(1,2,3,4,5,6,7)$ to
$(-2,-1,-5,-4,-3,6,7)$.) So $\rho(\boldsymbol{\delta})$ is the signed
permutation obtained from $\rho=(N-1,N-2,\dots,1,0)$ by dividing up the domain
$(1,2,\dots,N)$ into consecutive blocks of sizes
$k_{1},k_{2},\dots,k_{r},N-k$, and on the first $r$ blocks, it reverses then
negates the values of $\rho$. In our example, $\rho=(6,5,4,3,2,1,0)$ and
$\rho(\boldsymbol{\delta})=(-5,-6,-2,-3,-4,1,0)$.

Thus we have

$\rho(\boldsymbol{\delta})_{i}=-(N-1-k_{1}-\cdots-k_{j}+(i-k_{1}-\cdots-
k_{j-1}))=-N-i+1+2(k_{1}+\cdots+k_{j-1})+k_{j}$

for $k_{1}+\cdots+k_{j-1}<i\leq k_{1}+\cdots+k_{j}$, and for $k<i\leq N$,

$\rho(\boldsymbol{\delta})_{i}=N-i.$

For $1\leq i\leq k$, let $j$ be such that
$\lambda_{j}=\boldsymbol{\delta}_{i}$.

$\displaystyle
Y_{i}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
$\displaystyle=(T_{i}\cdots T_{N-1})(T_{N}\cdots T_{0})(T_{1}^{-1}\cdots
T_{i-1}^{-1})f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
(2.20) $\displaystyle=(T_{i}\cdots T_{N-1})(T_{N}\cdots
T_{0})t^{-(i-k_{1}-\cdots-
k_{j-1}-1)}f_{(\lambda_{j}\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}-1},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
(2.21) $\displaystyle=t^{-(i-k_{1}-\cdots-k_{j-1}-1)}(T_{i}\cdots
T_{N-1})(T_{N}\cdots
T_{1})q^{\boldsymbol{\delta}_{i}}f_{((-\lambda_{j})\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}-1},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
(2.22) $\displaystyle=q^{\boldsymbol{\delta}_{i}}t^{-(i-k_{1}-\cdots-
k_{j-1}-1)}(T_{i}\cdots
T_{N-1})T_{N}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}-1},\ldots,\lambda_{r}^{k_{r}},0^{N-k},(-\lambda_{j}))}$
(2.23) $\displaystyle=q^{\boldsymbol{\delta}_{i}}t^{-(i-k_{1}-\cdots-
k_{j-1}-1)}(T_{i}\cdots
T_{N-1})f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}-1},\ldots,\lambda_{r}^{k_{r}},0^{N-k},\lambda_{j})}$
(2.24) $\displaystyle=q^{\boldsymbol{\delta}_{i}}t^{-(i-k_{1}-\cdots-
k_{j-1}-1)}t^{k_{1}+\cdots+k_{j}-i}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{j}^{k_{j}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
$\displaystyle=q^{\boldsymbol{\delta}_{i}}t^{2(k_{1}+\cdots+k_{j-1})+k_{j}-2i+1}f_{\boldsymbol{\delta}},$

where (2.20), (2.22), and (2.24) are due to (2.16), (2.17), and the fact that
$-\boldsymbol{\delta}_{i}>0\geq\boldsymbol{\delta}_{\ell}$ for all $\ell$,
(2.21) is due to (2.14), and (2.23) is due to (2.19).

Since $\rho(\boldsymbol{\delta})_{i}=-N+1-i+2(k_{1}+\cdots+k_{j-1})+k_{j}$,
the exponent of $t$ is equal to $N-i+\rho(\boldsymbol{\delta})_{i}$, so
$Y_{i}f_{\boldsymbol{\delta}}=y_{i}(\boldsymbol{\delta})f_{\boldsymbol{\delta}}$
for $1\leq i\leq k$.

For $k+1\leq i\leq N$, we have

(2.25) $\displaystyle
Y_{i}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
$\displaystyle=(T_{i}\cdots T_{N-1})(T_{N}\cdots
T_{0})t^{-(i-1-k)}f_{(0,\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k-1})}$
(2.26) $\displaystyle=t^{k+1-i}(T_{i}\cdots T_{N-1})(T_{N}\cdots
T_{1})t_{0}f_{(0,\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k-1})}$
(2.27) $\displaystyle=t_{0}t^{k+1-i}(T_{i}\cdots
T_{N-1})T_{N}t^{N-k-1}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
(2.28) $\displaystyle=t_{0}t^{N-i}(T_{i}\cdots
T_{N-1})t_{N}f_{(\lambda_{1}^{k_{1}},\ldots,\lambda_{r}^{k_{r}},0^{N-k})}$
(2.29) $\displaystyle=t_{0}t_{N}t^{N-i}t^{N-i}f_{\boldsymbol{\delta}}$

where (2.25), (2.26), and (2.29) are due to (2.16) and (2.17), (2.26) is due
to (2.15), and (2.28) is due to (2.18).

For $k+1\leq i\leq N$, we have
$y_{i}(\boldsymbol{\delta})=t^{N-i+\rho(\boldsymbol{\delta})_{i}}(t_{0}t_{N})=t^{2N-2i}t_{0}t_{N}$.

∎

###  2.3. The monomials in the qKZ family $f_{\mu}(z_{1}\dots,z_{n};q,t)$

We define two partial orders on $\mathbb{Z}^{N}$.

######  Definition 2.8.

The _dominance order_ on compositions in $\mathbb{Z}^{N}$ is defined as
follows: $\mu\geq\nu$ if for $1\leq j\leq N$ we have
$\sum_{i=1}^{j}(\mu_{i}-\nu_{i})\geq 0$. We define a second order $\preceq$ on
compositions as follows. Let $\mu^{+}$ be the unique element in
$W_{0}\cdot\mu$ such that $\mu^{+}$ is a partition, i.e. such that
$\mu_{1}\geq\mu_{2}\geq\dots\geq 0$. Then we say that $\mu\preceq\nu$ if
$\mu^{+}<\nu^{+}$, or if $\mu^{+}=\nu^{+}$ and $\mu\leq\nu$.

For example, if $\mu=(-2,0)$, then the compositions $\nu\in\mathbb{Z}^{2}$
such that $\nu\preceq\mu$ are:

$(-2,0),(1,1),(1,-1),(-1,1),(-1,-1),(1,0),(-1,0),(0,1),(0,-1),(0,0).$

It is well-known [Sah99] (see also [Can17]) that the non-symmetric Koornwinder
polynomials $E_{\mu}(\mathbf{z})$ have the form

$E_{\mu}(\mathbf{z})=\mathbf{z}^{\mu}+\sum_{\nu\prec\mu}c_{\nu}\mathbf{z}^{\nu}.$

Moreover, it follows from the definitions

that the nonsymmetric Koornwinder polynomials $E_{\mu}$ and any qKZ family
$f_{\mu}$ are related via a triangular change of basis.

######  Proposition 2.9.

[CGdGW16, Proposition 1] Let $\\{f_{\mu}\\}$ be a qKZ family. Then the
nonsymmetric Koornwinder polynomials $E_{\mu}$ and the qKZ family $f_{\mu}$
are related via an invertible triangular change of basis:

$E_{\mu}=\sum_{\nu\preceq\mu}c_{\mu\nu}(q,t)f_{\nu},\text{ and
}f_{\mu}=\sum_{\nu\preceq\mu}d_{\mu\nu}(q,t)E_{\nu}$

for suitable rational coefficients $c_{\mu\nu}(q,t)$ and $d_{\mu\nu}(q,t)$.

It follows that the Laurent monomials appearing in the support of each
$f_{\mu}$ can be characterized as follows.

######  Corollary 2.10.

Let $\\{f_{\mu}\\}$ be a qKZ family. Then $f_{\mu}(\mathbf{z};q,t)$ has the
form

$f_{\mu}(\mathbf{z};q,t)=\sum_{\nu\preceq\mu}e_{\mu\nu}(q,t)\mathbf{z}^{\nu}.$

##  3\. The proof of Theorem 1.10

The main ingredient in our proof of Theorem 1.10 is the following theorem,
which says that the Laurent polynomials $\\{F_{\mu}(\mathbf{z};t)\\}$ are a
qKZ family. In what follows, if $\mu$ is a word in $\\{1,-1,0\\}^{N}$ then we
define

$||\mu||=N+r$, where $r$ is the number of $0$’s in $\mu$.

######  Theorem 3.1.

Let $\lambda\in\\{1,-1,0\\}^{N}$ be a composition. Then if we let $\mu$ range
over all signed permutations of $\lambda$, the collection
$\\{F_{\mu}(\mathbf{z};t)\\}_{\mu}\subset\mathbb{C}[z_{1}^{\pm
1},\dots,z_{N}^{\pm 1}]$ of Laurent polynomials is a qKZ family, in the sense
of Definition 2.4.

We will prove Theorem 3.1 over the course of this section. Recall that we
identify $\mathord{\bullet}=1$, $\mathord{\circ}=-1$, and $\mathord{\ast}=0$.
We start by recalling some “Matrix Ansatz”-style relations among the
generating polynomials $R(\mu)$ that previously appeared in [CMW17]. It will
turn out that the Matrix Ansatz relations are closely related to the action of
the Hecke operators.

######  Theorem 3.2 ([CMW17, Theorem 5.4]).

For any words $x$ and $y$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}=\\{-1,0,1\\}$,

(3.1) $\displaystyle tR(x\mathord{\bullet}\mathord{\circ}y)$
$\displaystyle=R(x\mathord{\circ}\mathord{\bullet}y)+\lambda_{||x||+||y||+2}(R(x\mathord{\bullet}y)+R(x\mathord{\circ}y))$
(3.2) $\displaystyle tR(x\mathord{\ast}\mathord{\circ}y)$
$\displaystyle=R(x\mathord{\circ}\mathord{\ast}y)+\lambda_{||x||+||y||+3}R(x\mathord{\ast}y)$
(3.3) $\displaystyle tR(x\mathord{\bullet}\mathord{\ast}y)$
$\displaystyle=R(x\mathord{\ast}\mathord{\bullet}y)+\lambda_{||x||+||y||+3}R(x\mathord{\ast}y)$
(3.4) $\displaystyle\beta R(x\mathord{\bullet})$ $\displaystyle=\delta
R(x\mathord{\circ})+\lambda_{||x||+1}R(x)$ (3.5) $\displaystyle\alpha
R(\mathord{\circ}x)$ $\displaystyle=\gamma
R(\mathord{\bullet}x)+\lambda_{||x||+1}R(x)$

where $\lambda_{N}=(\alpha\beta t^{N-1}-\gamma\delta)$.

We can rewrite Theorem 3.2 in terms of the $\tilde{R}(\mu)$, obtaining the
following statements.

######  Corollary 3.3.

For any words $x$ and $y$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}=\\{-1,0,1\\}$,

(3.6) $\displaystyle t\tilde{R}(x\mathord{\bullet}\mathord{\circ}y)$
$\displaystyle=\tilde{R}(x\mathord{\circ}\mathord{\bullet}y)+(t-1)(\tilde{R}(x\mathord{\bullet}y)+\tilde{R}(x\mathord{\circ}y))$
(3.7) $\displaystyle t\tilde{R}(x\mathord{\ast}\mathord{\circ}y)$
$\displaystyle=\tilde{R}(x\mathord{\circ}\mathord{\ast}y)+(t-1)\tilde{R}(x\mathord{\ast}y)$
(3.8) $\displaystyle t\tilde{R}(x\mathord{\bullet}\mathord{\ast}y)$
$\displaystyle=\tilde{R}(x\mathord{\ast}\mathord{\bullet}y)+(t-1)\tilde{R}(x\mathord{\ast}y)$
(3.9) $\displaystyle\alpha\tilde{R}(\mathord{\circ}x)$
$\displaystyle=\gamma\tilde{R}(\mathord{\bullet}x)+(t-1)\tilde{R}(x).$ (3.10)
$\displaystyle\beta\tilde{R}(x\mathord{\bullet})$
$\displaystyle=\delta\tilde{R}(x\mathord{\circ})+(t-1)\tilde{R}(x)$

The following lemmas, which are straightforward to prove, will be helpful for
working with the operators.

######  Lemma 3.4.

Let $1\leq i\leq N-1$, and suppose that $G$ is a polynomial that is
independent of $z_{i}$ and $z_{i+1}$. Then we have the following.

(3.11)
$\displaystyle\widetilde{T}_{i}\left((z_{i}-1)\left(\frac{1}{z_{i+1}}-1\right)G\right)$
$\displaystyle=(z_{i+1}-1)\left(\frac{1}{z_{i}}-1\right)G.$ (3.12)
$\displaystyle\widetilde{T}_{i}\left((z_{i}-1)G\right)$
$\displaystyle=\left((z_{i+1}-1)-(t-1)\right)G.$ (3.13)
$\displaystyle\widetilde{T}_{i}\left(\left(\frac{1}{z_{i+1}}-1\right)G\right)$
$\displaystyle=\left(\left(\frac{1}{z_{i}}-1\right)-(t-1)\right)G.$ (3.14)
$\displaystyle\widetilde{T}_{i}(G)$ $\displaystyle=tG.$

######  Lemma 3.5.

Suppose that $G$ is a polynomial that is independent of $z_{1}$. Then

(3.15) $\displaystyle q\widetilde{T}_{0}\left((z_{1}^{-1}-1)G\right)$
$\displaystyle=(z_{1}-1)G+\frac{1-t}{\gamma}G.$ (3.16)
$\displaystyle\widetilde{T}_{0}\left(G\right)$ $\displaystyle=t_{0}G.$

Suppose that $G$ is a polynomial that is independent of $z_{N}$. Then

(3.17) $\displaystyle\widetilde{T}_{N}\left((z_{N}-1)G\right)$
$\displaystyle=(z_{N}^{-1}-1)G+\frac{1-t}{\delta}G.$ (3.18)
$\displaystyle\widetilde{T}_{N}\left(G\right)$ $\displaystyle=t_{N}G$

###### Proof.

If $G$ is independent of $z_{1}$, then

$\frac{1-s_{0}}{z_{1}-qz_{1}^{-1}}G=0,\qquad\frac{1-s_{0}}{z_{1}-qz_{1}^{-1}}(z_{1}^{-1}-1)G=\frac{z_{1}^{-1}-q^{-1}z_{1}}{z_{1}-qz_{1}^{-1}}G=-q^{-1}G.$

We also have

$ac=-\frac{\alpha}{\gamma},\quad a+c=-\frac{1-t+\alpha-\gamma}{\gamma},\quad
bd=-\frac{\beta}{\delta},\quad b+d=-\frac{1-t+\beta-\delta}{\delta}.$

Then

$\widetilde{T}_{0}(G)=-acq^{-1}G=t_{0}G$

and

$\displaystyle q\widetilde{T}_{0}\left((z_{1}^{-1}-1)G\right)$
$\displaystyle=-ac(z_{1}^{-1}-1)G-(z_{1}-a-c+acz_{1}^{-1})(-G)$
$\displaystyle=z_{1}G+(ac-a-c)G=(z_{1}-1)+\frac{1-t}{\gamma}G.$

If $G$ is independent of $z_{N}$, then

$\frac{1-s_{N}}{z_{N}-z_{N}^{-1}}G=0,\qquad\frac{1-s_{N}}{z_{N}-z_{N}^{-1}}(z_{N}-1)G=\frac{z_{N}-z_{N}^{-1}}{z_{N}-z_{N}^{-1}}G=G.$

Then

$\widetilde{T}_{N}(G)=-bdG=t_{N}G$

and

$\displaystyle\widetilde{T}_{N}\left((z_{N}-1)G\right)$
$\displaystyle=-bd(z_{N}-1)G+(bdz_{N}-b-d+z_{N}^{-1})G$
$\displaystyle=z_{N}^{-1}G+(bd-b-d)G=(z_{N}^{-1}-1)G+\frac{1-t}{\delta}G.$

∎

Our first goal is to prove the following.

######  Theorem 3.6.

Choose $1\leq i\leq N-1$. For any words $x$ and $y$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$, where $|x|=i-1$, we
have that

$\widetilde{T}_{i}(F_{x\mathord{\bullet}\mathord{\circ}y}(\mathbf{z};t))=F_{x\mathord{\circ}\mathord{\bullet}y}(\mathbf{z};t).$

###### Proof.

Let $\mu=(\mu_{1},\dots,\mu_{N})=x\mathord{\bullet}\mathord{\circ}y$ and
$|x|=i-1$, so that the $\mathord{\bullet}$ and $\mathord{\circ}$ are in
positions $i$ and $i+1$. As before, let $V=\\{i\ |\ \mu_{i}\in\\{\pm 1\\}\\}$.

In the definition of $F_{\mu}(\mathbf{z};t)$, we will divide the sum over
subsets of $[N]$ into four cases based on whether the subset contains both $i$
and $i+1$, just one of them, or neither. We then get:

$\displaystyle F_{x\mathord{\bullet}\mathord{\circ}y}(\mathbf{z};t)$
$\displaystyle=$
$\displaystyle(z_{i}-1)\left(\frac{1}{z_{i+1}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))$
$\displaystyle+(z_{i}-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{i\in
S}(z_{i}^{\mu_{i}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle+\left(\frac{1}{z_{i+1}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}\mathord{\circ}(y|_{\overline{S}})).$

Using (3.11), we get

$\displaystyle\widetilde{T}_{i}$
$\displaystyle\left((z_{i}-1)\left(\frac{1}{z_{i+1}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))\right)=$
$\displaystyle(z_{i+1}-1)\left(\frac{1}{z_{i}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}})).$

Using (3.12), we get

$\displaystyle\widetilde{T}_{i}\left((z_{i}-1)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}})\right)=$
$\displaystyle(z_{i+1}-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle-(t-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}})).$

Using (3.13), we get

$\displaystyle\widetilde{T}_{i}\left(\left(\frac{1}{z_{i+1}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))\right)=$
$\displaystyle\left(\frac{1}{z_{i}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))$
$\displaystyle-(t-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}})).$

Using (3.14), we get

$\displaystyle\widetilde{T}_{i}\left(\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(\mu|_{\overline{S}})\right)=t\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(\mu|_{\overline{S}}).$

Therefore

$\displaystyle\widetilde{T}_{i}F_{x\mathord{\bullet}\mathord{\circ}y}(\mathbf{z};t)$
$\displaystyle=$
$\displaystyle(z_{i+1}-1)\left(\frac{1}{z_{i}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))$
$\displaystyle+(z_{i+1}-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle+\left(\frac{1}{z_{i}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)(t\tilde{R}((x|_{\overline{S}})\mathord{\bullet}\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle-(t-1)(\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))+\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}}))).$

Now we use (3.6)

and get that

$t\tilde{R}((x|_{\overline{S}})\mathord{\bullet}\mathord{\circ}(y|_{\overline{S}}))-(t-1)(\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))+\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}})))=\tilde{R}((x|_{\overline{S}})\mathord{\circ}\mathord{\bullet}(y|_{\overline{S}})).$

So

$\displaystyle\widetilde{T}_{i}F_{x\mathord{\bullet}\mathord{\circ}y}(\mathbf{z};t)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{z_{i}}-1\right)(z_{i+1}-1)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))$
$\displaystyle+(z_{i+1}-1)\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle+\left(\frac{1}{z_{i}}-1\right)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}\mathord{\bullet}(y|_{\overline{S}}))$
$\displaystyle=$ $\displaystyle
F_{x\mathord{\circ}\mathord{\bullet}y}(\mathbf{z};t),$

as desired. ∎

We also need to prove the following identities.

######  Theorem 3.7.

Choose $1\leq i\leq N-1$. For any words $x$ and $y$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$, where $|x|=i-1$, we
have that

(3.19)
$\widetilde{T}_{i}(F_{x\mathord{\ast}\mathord{\circ}y}(\mathbf{z};t))=F_{x\mathord{\circ}\mathord{\ast}y}(\mathbf{z};t)$

and

(3.20)
$\widetilde{T}_{i}(F_{x\mathord{\bullet}\mathord{\ast}y}(\mathbf{z};t))=F_{x\mathord{\ast}\mathord{\bullet}y}(\mathbf{z};t).$

###### Proof.

We prove (3.19). Let
$\mu=(\mu_{1},\dots,\mu_{N})=x\mathord{\ast}\mathord{\circ}y$ and $|x|=i-1$,
so that the $\mathord{\ast}$ and $\mathord{\circ}$ are in positions $i$ and
$i+1$. As before, let $V=\\{i\ |\ \mu_{i}\in\\{\pm 1\\}\\}$.

We divide the sum in the definition of $F_{\mu}(\mathbf{z};t)$ over subsets of
$[N]$ into two cases based on whether the subset contains $i+1$ or not. We
then get:

$\displaystyle F_{x\mathord{\ast}\mathord{\circ}y}(\mathbf{z};t)$
$\displaystyle=$ $\displaystyle\left(z_{i+1}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}\mathord{\circ}(y|_{\overline{S}})).$

Applying (3.13) and (3.14) to the two sums in
$\widetilde{T}_{i}(F_{\mu}(\mathbf{z};t))$, we get

$\displaystyle\widetilde{T}_{i}(F_{x\mathord{\ast}\mathord{\circ}y}(\mathbf{z};t))=$
$\displaystyle\left(z_{i}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))$
$\displaystyle-(t-1)\sum_{S\subseteq V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))$
$\displaystyle+t\sum_{S\subseteq V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}\mathord{\circ}(y|_{\overline{S}}))$
$\displaystyle=$ $\displaystyle\left(z_{i}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))$
$\displaystyle\quad+\sum_{S\subseteq V\setminus\\{i+1\\}}$
$\displaystyle\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)(t\tilde{R}((x|_{\overline{S}})\mathord{\ast}\mathord{\circ}(y|_{\overline{S}}))-(t-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))).$

Using (3.7), we get

$\displaystyle\widetilde{T}_{i}(F_{x\mathord{\ast}\mathord{\circ}y}(\mathbf{z};t))=$
$\displaystyle\left(z_{i}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\ast}(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\circ}\mathord{\ast}(y|_{\overline{S}})),$

which is equal to $F_{x\mathord{\circ}\mathord{\ast}y}(\mathbf{z};t)$ when
written as a sum of the two cases depending on whether or not $i$ is in $S$.

(3.20) is proved in the same manner. ∎

######  Theorem 3.8.

For any word $x$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$ of length $|x|=N-1$,
we have that

$q\widetilde{T}_{0}(F_{\mathord{\circ}x}(\mathbf{z};t))=F_{\mathord{\bullet}x}(\mathbf{z};t).$

###### Proof.

Let $\mu=(\mu_{1},\dots,\mu_{N})=\mathord{\circ}x$ and $|x|=N-1$, so that
$\mathord{\circ}$ is in position $1$. As before, let $V=\\{i\ |\
\mu_{i}\in\\{\pm 1\\}\\}$.

We divide the sum over subsets of $[N]$ in the definition of
$F_{\mu}(\mathbf{z};t)$ into two cases based on whether the subset contains
$1$ or not. We get:

$\displaystyle F_{\mathord{\circ}x}(\mathbf{z};t)$ $\displaystyle=$
$\displaystyle\left(z_{1}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\sum_{S\subseteq V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(\mathord{\circ}(x|_{\overline{S}})).$

Thus

$\displaystyle q\widetilde{T}_{0}(F_{\mathord{\circ}x}(\mathbf{z};t))$
$\displaystyle=$ $\displaystyle\left(z_{1}-1\right)\sum_{S\subseteq
V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\frac{1-t}{\gamma}\sum_{S\subseteq
V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\frac{\alpha}{\gamma}\sum_{S\subseteq
V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(\mathord{\circ}(x|_{\overline{S}})).$

Now we use (3.9) to get that

$\tilde{R}(\mathord{\bullet}(x|\overline{S}))=\frac{1}{\gamma}\left(\alpha\tilde{R}(\mathord{\circ}(x|_{\overline{S}}))+(1-t)\tilde{R}(x|_{\overline{S}})\right).$

Then

$\displaystyle q\widetilde{T}_{0}(F_{\mathord{\circ}x}(\mathbf{z};t))$
$\displaystyle=$ $\displaystyle\left(z_{1}-1\right)\sum_{S\subseteq
V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\sum_{S\subseteq V\setminus\\{1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})\left(\frac{1-t}{\gamma}+\frac{\alpha}{\gamma}\right)$
$\displaystyle=$ $\displaystyle F_{\mathord{\bullet}x}(\mathbf{z};t).$

∎

######  Theorem 3.9.

For any word $x$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$ of length $|x|=N-1$,
we have that

$\widetilde{T}_{N}(F_{x\mathord{\bullet}}(\mathbf{z};t))=F_{x\mathord{\circ}}(\mathbf{z};t).$

###### Proof.

Let $\mu=(\mu_{1},\dots,\mu_{N})=x\mathord{\bullet}$ and $|x|=N-1$, so that
$\mathord{\bullet}$ is in position $N$. As before, let $V=\\{i\ |\
\mu_{i}\in\\{\pm 1\\}\\}$.

We divide the sum over subsets of $[N]$ in the definition of
$F_{\mu}(\mathbf{z};t)$ into two cases based on whether the subset contains
$N$ or not. We get:

$\displaystyle F_{x\mathord{\bullet}}(\mathbf{z};t)$ $\displaystyle=$
$\displaystyle\left(z_{N}-1\right)\sum_{S\subseteq
V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\sum_{S\subseteq V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})\mathord{\bullet}).$

Thus

$\displaystyle\widetilde{T}_{N}(F_{x\mathord{\bullet}}(\mathbf{z};t))$
$\displaystyle=$ $\displaystyle\left(z_{N}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\frac{1-t}{\delta}\sum_{S\subseteq
V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\frac{\beta}{\delta}\sum_{S\subseteq
V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}}).$

Now we use (3.10) to get that

$\tilde{R}(x|_{\overline{S}})=\frac{1}{\delta}\left(\beta\tilde{R}((x|_{\overline{S}})\mathord{\bullet})+(1-t)\tilde{R}(x|_{\overline{S}})\right).$

Then

$\displaystyle\widetilde{T}_{N}(F_{x\mathord{\bullet}}(\mathbf{z};t))$
$\displaystyle=$ $\displaystyle\left(z_{N}^{-1}-1\right)\sum_{S\subseteq
V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})$
$\displaystyle+\sum_{S\subseteq V\setminus\\{N\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}(x|_{\overline{S}})\left(\frac{1-t}{\delta}+\frac{\beta}{\delta}\right)$
$\displaystyle=$ $\displaystyle F_{x\mathord{\circ}}(\mathbf{z};t).$

∎

We also prove the following identity.

######  Theorem 3.10.

Choose words $x$ and $y$ in the letters
$\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$ where $|x|=i-1$ and
additionally choose
$a\in\\{\mathord{\circ},\mathord{\ast},\mathord{\bullet}\\}$. Then

(3.21) $\widetilde{T}_{i}(F_{xaay}(\mathbf{z};t))=tF_{xaay}(\mathbf{z};t).$

###### Proof.

Let $\mu=(\mu_{1},\ldots,\mu_{N})=xaay$. If $a=\mathord{\ast}$, the equality
trivially holds by (3.14). Otherwise, we will show $F_{xaay}(\mathbf{z};t)$ is
symmetric in variables $z_{i},z_{i+1}$. Let $u=\mu_{i}=\mu_{i+1}$. Then

$\displaystyle F_{xaay}(\mathbf{z};t)$ $\displaystyle=$
$\displaystyle(z_{i}^{u}-1)(z_{i+1}^{u}-1)\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})(y|_{\overline{S}}))$
$\displaystyle+((z_{i}^{u}-1)+(z_{i+1}^{u}-1))\sum_{S\subseteq
V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})a(y|_{\overline{S}}))$
$\displaystyle+\sum_{S\subseteq V\setminus\\{i,i+1\\}}\prod_{j\in
S}(z_{j}^{\mu_{j}}-1)\tilde{R}((x|_{\overline{S}})aa(y|_{\overline{S}})).$

Thus $F_{xaay}(\mathbf{z};t)$ is symmetric in $z_{i},z_{i+1}$, so
$\frac{1-s_{i}}{z_{i}-z_{i+1}}F_{xaay}(\mathbf{z};t)=0$, from which (3.21)
follows. ∎

Finally we are ready to put together our previous results to prove Theorem
3.1.

###### Proof.

First note that Theorem 3.8 proves (2.14). It follows from (1.3) that for any
$\mu$ with $\mu_{i}=\mathord{\ast}$, $\tilde{R}(\mu)$ is independent of
$z_{i}$. Hence (3.16) implies (2.15) and (3.18) implies (2.18). Finally we
have that Theorem 3.10 proves (2.16), Theorems 3.6 and 3.7 proves (2.17), and
Theorem 3.9 proves (2.19). ∎

Now we can put all these ingredients together to complete the proof of the
main theorem.

###### Proof of Theorem 1.10.

By Theorem 3.1, the $\\{F_{\mu}(\mathbf{z};t)\\}_{\mu}$ are a qKZ family and
that
$Y_{i}F_{\boldsymbol{\delta}}=y_{i}(\boldsymbol{\delta})F_{\boldsymbol{\delta}}$
for $i=1,\dots,N$. Together with Definition 2.3 and the fact that the leading
coefficient of $F_{\boldsymbol{\delta}}$ is $\mathbf{z}^{\boldsymbol{\delta}}$
(see Remark 1.12), this implies that $F_{\boldsymbol{\delta}}$ equals the
nonsymmetric Koornwinder polynomial $E_{\boldsymbol{\delta}}$. The theorem now
follows from Proposition 2.5. ∎

###### Proof of Theorem 1.15.

The proof of Theorem 1.15 follows from the Theorem 1.10.

$\displaystyle K_{\lambda}(\mathbf{z};t)$ $\displaystyle=$
$\displaystyle\sum_{\mu\in W_{0}(\lambda)}F_{\mu}(\mathbf{z};t)$
$\displaystyle=$ $\displaystyle\sum_{\mu\in W_{0}(\lambda)}\
\sum_{S\subseteq[N]}\tilde{R}(\mu|_{\overline{S}})\prod_{i\in
S}(z_{i}^{\mu_{i}}-1)$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{N-r}\
\sum_{S\in{[N]\choose k}}\ \ \sum_{\nu\in
W_{0}(1^{N-r-k},0^{r})}\tilde{R}(\nu)\cdot\prod_{i\in
S}\left((z_{i}-1)+(z_{i}^{-1}-1)\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-r}\tilde{Z}_{N-k-r,r}\cdot
e_{k}(z_{1}+z_{1}^{-1}-2,\ldots,z_{N}+z_{N}^{-1}-2).$

where $W_{0}$ is the set of signed permutations of the appropriate
cardinality, so e.g. the sum over $\nu\in W_{0}(1^{N-r-k},0^{r})$ indicates
that we are summing over all signed permutations of the word
$(1^{N-r-k},0^{r})$.

∎

##  4\. Conclusion

In this work we have given a combinatorial formula for the Koornwinder
polynomials $K_{\lambda}$ associated to partitions $\lambda=(1^{N-r},0^{r})$
by using the combinatorics of the open boundary two-species ASEP. This work
naturally leads to several open problems which we hope to pursue in future
work.

A first problem is to find a vertex model analogue of our rhombic staircase
tableaux, corresponding to particle models with open boundaries and their
associated type $\tilde{C}$ special functions. We note that there has been a
great deal of recent work connecting particle models on the ring and (type
$A$) special functions to _vertex models_ , see [BW19, ABW21, ANP]. In
particular, [CdGW15, Section 6] gave a lattice model interpretation of their
matrix product formula for Macdonald polynomials. Subsequently, [ANP] gave a
queue vertex model formula for Macdonald polynomials, which encodes the
multiline queues of Martin [Mar20], and gives new proofs of some formulas for
the stationary distribution of the multispecies ASEP on a ring using the
Yang–Baxter equation.

A second problem is to extend our results to give a combinatorial formula for
all Koornwinder polynomials. Towards that end, we now briefly sketch how the
analogous problem was solved for type $A$ Macdonald polynomials, and what are
the difficulties in the Koornwinder case.

Let us refer to the multispecies ASEP on a ring involving particles
$\\{0,1,\dots,m\\}$ as the _rank $m$_ multispecies ASEP on a ring. If $m=1$,
the multispecies ASEP on a ring is rather trivial – its stationary
distribution is uniform – but for $m>1$, the stationary distribution becomes
more interesting. To get some insight on what happens for $m>1$, recall that
the multispecies ASEP on a ring is connected to the type $A$ Macdonald
polynomials. More specifically, the partition function of the rank $m$
multispecies ASEP on a ring is the specialization of a related Macdonald
polynomial (with largest part $m$) at $x_{1}=\cdots=x_{N}=1$ and $q=1$
[CdGW15]. Moreover, when $q=1$, Macdonald polynomials admit the following
factorization [Mac95, Chapter VI, Equation (4.14)(vi)]:

(4.1) $P_{\lambda}(\mathbf{x};1,t)=\prod_{i\geq
1}P_{(1^{\lambda^{\prime}_{i}},0^{N-\lambda^{\prime}_{i}})}(\mathbf{x};1,t)=\prod_{i\geq
1}P_{(1^{\lambda^{\prime}_{i}})}(\mathbf{x};1,t)=\prod_{i\geq
1}e_{{\lambda^{\prime}_{i}}}(\mathbf{x}),$

where $e_{\mu}$ is the elementary symmetric polynomial. This means that the
partition function of the rank $m$ mASEP on a ring is just the product of some
elementary symmetric polynomials. Martin’s multiline queue formula for the
stationary distribution of the mASEP [Mar20] reflects this structure: each
multiline queue is built by stacking rows of balls on top of each other, where
each row of balls can be thought of as describing a rank $1$ ASEP. Once one
has the multiline queues with the $t$ statistic, adding the parameters $q$ and
$x_{i}$ to obtain a formula for Macdonald polynomials turns out to be
miraculously simple [CMW22].

Returning to the ASEP with open boundaries, let us refer to the multispecies
ASEP with particles $\\{0,\pm 1,\pm 2,\ldots,\pm m\\}$ as the _rank $m$_ open
boundary ASEP. Unlike the ASEP on a ring, the rank $1$ open boundary ASEP has
a very non-trivial stationary distribution. Even in the case of the rank $1$
open ASEP with no second class particles, there was a tremendous amount of
work on the ASEP [DEHP93, USW04, BE04, DS05, BCE+06, CW07] before a
combinatorial formula for the stationary distribution (with all parameters
$\alpha,\beta,\gamma,\delta,q$ general) was given in [CW11]; this formula was
subsequently extended to the rank $1$ case with second class particles in
[CMW17], using rhombic staircase tableaux. Similar to the case of the ASEP on
a ring, the partition function of the rank $m$ open boundary mASEP is the
specialization of a related Koornwinder polynomial (with largest part $m$) at
$x_{1}=\cdots=x_{N}=1$ and $q=1$ [CGdGW16]. Moreover, just as Macdonald
polynomials factor at $q=1$, the Koornwinder polynomials admit the
factorization from 1.16:

(4.2) $K_{\lambda}(\mathbf{z};a,b,c,d;q,t)=\prod_{i\geq
1}K_{(1^{\lambda^{\prime}_{i}},0^{N-\lambda^{\prime}_{i}})}(\mathbf{z};a,b,c,d;q,t)=\prod_{i\geq
1}K_{1^{\lambda^{\prime}_{i}}}(\mathbf{z};a,b,c,d;q,t).$

In this paper we have given formulas (Theorem 1.10 and Theorem 1.15) in terms
of rhombic staircase tableaux for the “rank 1” Koornwinder polynomials
$K_{\lambda}$ appearing on the right-hand side of (4.2). Using the
factorization (4.2), we also gave a combinatorial formula for arbitrary
Koornwinder polynomials at $q=1$ (Corollary 1.17). Thus, one might hope to
figure out how to insert the parameter $q$ into Corollary 1.17, so as to give
a formula for general Koornwinder polynomials. In particular, by analogy with
the Macdonald case, one might hope that there might be a combinatorial formula
for arbitrary Koornwinder polynomials where the new combinatorial object is
somehow built by stacking rhombic staircase tableaux on top of each other.
However, as rhombic staircase tableaux are already quite complicated, and our
formulas in Theorem 1.10 and Theorem 1.15 are not that simple, it is so far
unclear to us how to define the correct object.

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