# Orthogonal polynomial duality and unitary symmetries of multi–species
ASEP$(q,\boldsymbol{\theta})$ and higher–spin vertex models via ∗–bialgebra
structure of higher rank quantum groups

Chiara Franceschini  University of Modena and Reggio Emilia, FIM, via G. Campi
213/b, 41125 Modena, Italy  Jeffrey Kuan  Texas A&M University, Department of
Mathematics, College Station TX, 77843-3368.  Zhengye Zhou  Texas A&M
University, Department of Mathematics, College Station TX, 77843-3368.

###### Abstract

We propose a novel, general method to produce orthogonal polynomial dualities
from the ∗–bialgebra structure of Drinfeld–Jimbo quantum groups. The
∗–structure allows for the construction certain unitary symmetries, which
imply the orthogonality of the duality functions. In the case of the quantum
group $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$, the result is a nested
multivariate $q$–Krawtchouk duality for the $n$–species
ASEP$(q,\boldsymbol{\theta})$. The method also applies to other quantized
simple Lie algebras and to stochastic vertex models.

As a probabilistic application of the duality relation found, we provide the
explicit formula of the $q-$shifted factorial moments (namely the $q$-analogue
of the Pochhammer symbol) for the two–species $q$–TAZRP (totally asymmetric
zero range process).

##  1 Introduction

Over the last several years, there has been a flurry of interest in so–called
multi–species models, also known as colored or multi–class models [BS15b,
BS15a, BS18, BGW, Gal, Kua17, Kua18a, KMMO16, Tak, Zho]. In the context of
interacting particle systems, these were first introduced in 1976 by [Lig76].
More recently, algebraic methods have been developed to find Markov duality
for these multi–species models [BS,BS2,BS3,Kuan-IMRN,KuanCMP], generalizing
Schütz’s method in [Sch97]. In these papers, the duality functions are
“triangular” duality functions, in the sense that when viewed as matrices they
are lower triangular matrices. However, a priori there is not necessarily a
reason to believe that triangular duality functions can be used for
probabilistic applications, although at times they are [CST16, Lin20] in the
context of KPZ universality class and for a single species only.

In some recent works by [FGG, CFGGR19, Gro19, CFG21], algebraic methods have
been developed to find orthogonal polynomial duality functions for interacting
particle systems, both in the symmetric and asymmetric cases. Regarding
applications of such orthogonal duality functions for symmetric particle
systems we mention [FRS] where they use orthogonal dualities in an open system
to infer information and partially characterize the non-equilibrium stationary
measure. In [ACR18] and [ACR21] the authors first present a quantitative
Boltzmann–Gibbs principle used to study macroscopic fluctuations and then they
study scaling limits of higher order fluctuation fields. In all the above it
was crucial to have duality functions which are polynomials orthogonal with
respect to the reversible measure of the Markov process considered. These
orthogonal polynomial duality functions are advantageous, from a probabilistic
application, because not only one can take advantage of the duality relation
but also of the orthogonal relation and the fact that observables of interest
can be written in terms of such orthogonal duality functions. In a sense, the
orthogonality guarantees that the duality function is “useful”, as long as the
dual process can be analyzed.

In general, having a duality relation for a particle system yields a dual
process with finitely many and, in particular, very few particles. Via the
analysis of these few dual particles one can gather information on some
relevant quantities for the initial process. These quantities (e.g. truncated
correlations for symmetric particle systems or $q^{-2}$ exponential moments
for asymmetric ones) are suggested by the type of duality function one has.

In this paper, we introduce a general method to produce orthogonal polynomial
dualities, using the ∗–structure of bialgebras. The ∗–structure allows for the
construction of unitary symmetries, from which the orthogonality of the
duality functions will follow. This method is used explicitly in the case of
the Drinfeld–Jimbo quantum group $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$, which
is an underlying algebraic structure of the $n$–species
ASEP$(q,\boldsymbol{\theta})$. As far as the authors know, this is the first
probabilistic application of ∗–bialgebras. This method is expected to work for
other simple Lie algebras (see upcoming work of the second and third authors).
Note that this method is not the same as the “inner product method” of
[CFGGR19] used to construct new duality functions starting from known ones and
which turns out to not generalize to higher–rank Lie algebras.

The $n$–species ASEP$(q,\boldsymbol{\theta})$ allows for $\theta^{x}$
particles to occupy site $x$, and has an asymmetry parameter of $q^{\theta}$.
In the $\theta\rightarrow\infty$ limit the process converges to a $n$–species
$q$–TAZRP (totally asymmetric zero range process), in which arbitrarily many
particles may occupy a site and all jumps occur in one direction. Applying a
charge–parity symmetry, the duality function converges to a nontrivial
$q$–Pochhammer. Furthermore, in this limit, the reversible measures converge
to a deterministic initial condition, since the totally asymmetric process is
not reversible. This suggests, a priori, that the duality can be used to find
explicit formulas for the $n$–species $q$–TAZRP. Indeed, for the $2$–species
$q$–TAZRP, we use the polynomial duality to give exact formulas for the
$q$–shifted factorial moments that can be used to yield the rate of
decorrelation at two points. This uses that the process is integrable, in the
sense that its generator satisfies the Yang–Baxter equation, and has explicit
transition probabilities expressed in terms of contour integral formulas.
Indeed, an analysis of the dual (finite–particle) process involves explicit
contour integral formulas for certain observables of the multi–species
$q$–TAZRP, very recently found by the second author in [Kua21]

The multi–species $q$–TAZRP is itself a special case of the multi–species
$q$–Hahn TAZRP, which is itself a special case of the multi–species higher
spin stochastic vertex model. Using the general algebraic setup of [Kua18a],
the orthogonality polynomial duality function for the
ASEP$(q,\boldsymbol{\theta})$ is also a duality function for the vertex model,
after applying the charge–parity symmetry (CP symmetry). See Figure 1 shows
all the processes discussed in the paper. The
$\boldsymbol{\theta}\rightarrow\infty$ limit corresponds to the
$\mu\rightarrow 0$ limit, so thus an explicit proof is additionally needed for
the $q$–Hahn TAZRP.

Figure 1: The various degenerations and limits explicitly mentioned in this
paper. See [Kua18a] for some more degenerations not mentioned in this paper.

The outline of the paper is as follows. Section 2 defines notation, formulas,
and we introduce all the models in a close boundary setting. Section 3 states
the main results and it ends with an application of one of the duality
relations shown. The proofs of all results are in section 4. Note that the
method of the proof generalizes to other Lie algebras, as discussed in Remark
6.

Acknowledgments. C.F. acknowledges the NSF grant DMS-1928930 which supported
her participation at the Mathematical Sciences Research Institute in Berkeley,
California, during the Fall 2021 semester. J.K. and Z.Z. would like to
acknowledge the Simons Foundation Collaboration Grant 713614, for funding
travel to Berkeley to visit C.F.

##  2 Definitions

###  2.1 $q$–notation

For any $q$ which is not a root of unity, define the $q$–deformed integer,
factorial, binomial, and multinomial by

$[n]_{q}=\frac{q^{n}-q^{-n}}{q-q^{-1}},\quad[n]_{q}^{!}=[1]_{q}\cdots[n]_{q},\quad\binom{n}{k}_{q}=\frac{[n]_{q}^{!}}{[k]_{q}^{!}[n-k]_{q}^{!}},\quad\binom{n}{k_{1},\ldots,k_{l}}_{q}=\frac{[n]_{q}^{!}}{[k_{1}]_{q}^{!}\cdots[k_{l}]_{q}^{!}}$

and the $q$–Pochhammer

$(a;q)_{n}=(1-a)(1-aq)\cdots\left(1-aq^{n-1}\right),\quad(a;q)_{0}:=1.$ (1)

A second $q$–deformed integer and factorial that will be used is

$\\{n\\}_{q^{2}}=\frac{1-q^{2n}}{1-q^{2}},\quad\\{n\\}_{q^{2}}=\\{1\\}_{q^{2}}\cdots\\{n\\}_{q^{2}}.$
(2)

The two $q$–deformations are related by

$q^{n-1}[n]_{q}=\\{n\\}_{q^{2}},\quad
q^{n(n-1)/2}[n]_{q}^{!}=\\{n\\}_{q^{2}}^{!}.$ (3)

The first $q$–deformation with the square brackets is more natural in an
algebraic setting, because it reflects the $q\mapsto q^{-1}$ symmetry, while
the second $q$–deformation with the curly brackets is more natural in an
probabilistic setting, because $q^{2}$ is the asymmetry parameter in the
asymmetric simple exclusion process.

Also, define the $q$–exponentials

$e_{q}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{(q;q)_{n}}=\frac{1}{(z;q)_{\infty}},\quad|z|<1,\quad\text{
and
}\quad\mathcal{E}_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}z^{n}}{(q;q)_{n}}=(-z;q)_{\infty}.$
(4)

The more common notation for $\mathcal{E}_{q}$ is $E_{q}$, but in this paper
the $\mathcal{E}$ will be used to avoid notation conflict with the quantum
group generators $E_{ij}$. The $q$–exponentials satisfy two
$q$–Baker–Campbell–Hausdorff equations, the first being

$\mathcal{E}_{q}(\lambda X)Ye_{q}\left(-\lambda
q^{\alpha}X\right)=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{(q;q)_{n}}[X,Y]_{n},$
(5)

where

$[X,Y]_{0}=Y,\quad[X,Y]_{n+1}=q^{n}X[X,Y]_{n}-q^{\alpha}[X,Y]_{n}X,\quad
n=1,2,\ldots,$ (6)

and the second being

$e_{q}(\lambda X)Y\mathcal{E}_{q}(-\lambda
X)=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{(q;q)_{n}}[X,Y]_{n}^{\prime},$ (7)

where

$\qquad[X,Y]_{0}^{\prime}=Y,\quad[X,Y]_{n+1}^{\prime}=X[X,Y]_{n}^{\prime}-q^{n}[X,Y]_{n}^{\prime}X,\quad
n=0,1,2,\ldots.$

If $xy=q^{2}yx$ then

$\mathcal{E}_{q^{2}}(x+y)=\mathcal{E}_{q^{2}}(x)\cdot\mathcal{E}_{q^{2}}(y)\quad\text{
and }\quad e_{q^{2}}(x+y)=e_{q^{2}}(y)\cdot e_{q^{2}}(x).$ (8)

The $e_{q^{2}}$ and $\mathcal{E}_{q^{2}}$ are mutual inverses of each other,
in the sense that

$e_{q^{2}}(z)\mathcal{E}_{q^{2}}(-z)=1.$ (9)

###  2.2 Orthogonal Polynomials

Recall the $q$–hypergeometric series

${}_{2}\varphi_{1}\left(\begin{array}[]{c}a,b\\\
c\end{array};q,z\right):=\sum_{k=0}^{\infty}\frac{(a;q)_{k}(b;q)_{k}}{(c;q)_{k}}\frac{z^{k}}{(q;q)_{k}},$
(10)

Note that for $a=q^{n}$ and $n\in\mathbb{N}$, the series is a polynomial of
degree $n$ with respect to the variable $b$. Now define the $q$–Krawtchouk
polynomial as in Chapter 14.14 of [KLS]

$K_{n}\left(q^{-x};p,c;q\right):={}_{2}\varphi_{1}\left(\begin{array}[]{c}q^{-x},q^{-n}\\\
q^{-c}\end{array};q,pq^{n+1}\right).$ (11)

These are orthogonal in the following sense: when $pq^{c}>1$ and
$c\in\mathbb{N}$,

$\sum_{x=0}^{c}\frac{(pq;q)_{c-x}(-1)^{c-x}}{(q;q)_{x}(q;q)_{c-x}}q^{\left(\frac{x}{2}\right)}\cdot
K_{m}\left(q^{-x};p,c;q\right)\cdot K_{n}\left(q^{-x};p,c;q\right)\\\
=\frac{(-1)^{n}p^{c}(q;q)_{c-n}(q;q)_{n}(pq;q)_{n}}{\left((q;q)_{c}\right)^{2}}\cdot
q^{\binom{c+1}{2}-\binom{n+1}{2}+cn}\cdot\delta_{m,n}.$

###  2.3 Quantum Groups and Representations

The Drinfeld–Jimbo quantum group $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$ is the
Hopf algebra with generators

$\\{E_{i,i+1},E_{i+1,i}:0\leq i<n\\},\\{q^{E_{ii}}:0\leq i\leq n\\}$

and relations

$\displaystyle q^{E_{ii}}q^{E_{jj}}=q^{E_{jj}}q^{E_{ii}}=q^{E_{ii}+E_{jj}}$
$\displaystyle\qquad\left[E_{i,i+1},E_{i+1,i}\right]=\frac{q^{E_{ii}-E_{i+1,i+1}}-q^{E_{i+1,i+1}-E_{ii}}}{q-q^{-1}}\quad\left[E_{i,i+1}E_{j+1,j}\right]=0,\quad
i\neq j$ $\displaystyle q^{E_{ii}}E_{i,i+1}=qE_{i,i+1}q^{E_{ii}}\quad
q^{E_{ii}}E_{i-1,i}=q^{-1}E_{i-1,i}q^{E_{ii}}\quad\left[q^{E_{ii}},E_{j,j+1}\right]=0,\quad
j\neq i,i-1$ $\displaystyle q^{E_{ii}}E_{i,i-1}=qE_{i,i-1}q^{E_{ii}}\quad
q^{E_{ii}}E_{i+1,i}=q^{-1}E_{i+1,i}q^{E_{ii}}\quad\left[q^{E_{ii}},E_{j,j-1}\right]=0,\quad
j\neq i,i+1$ $\displaystyle
E_{i,i+1}^{2}E_{j,j+1}-\left(q+q^{-1}\right)E_{i,i+1}E_{j,j+1}E_{i,i+1}+E_{j,j+1}E_{i,i+1}^{2}=0,\quad
i=j\pm 1$ $\displaystyle
E_{i,i-1}^{2}E_{j,j-1}-\left(q+q^{-1}\right)E_{i,i-1}E_{j,j-1}E_{i,i-1}+E_{j,j-1}E_{i,i-1}^{2}=0,\quad
i=j\pm 1$
$\displaystyle{\left[E_{i,i+1},E_{j,j+1}\right]=0=\left[E_{i,i-1},E_{j,j-1}\right],\quad
i\neq j\pm 1}$

The co–product is defined by

$\displaystyle{}\Delta\left(E_{i,i+1}\right)$ $\displaystyle=E_{i,i+1}\otimes
1+q^{E_{ii}-E_{i+1,i+1}}\otimes E_{i,i+1}$
$\displaystyle\Delta\left(E_{i+1,i}\right)$ $\displaystyle=1\otimes
E_{i+1,i}+E_{i+1,i}\otimes q^{-(E_{ii}-E_{i+1,i+1})}$
$\displaystyle\Delta\left(q^{E_{i,i}}\right)$
$\displaystyle=q^{E_{i,i}}\otimes q^{E_{i,i}}$

The antipode will not be explicitly needed.

When $q$ is a nonzero real number, then $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$
has the structure of a Hopf $*$–algebra with the $*$ involution defined by

$E_{i,i+1}^{*}=E_{i+1,i}q^{E_{ii}-E_{i+1,i+1}},\quad
E_{i+1,i}^{*}=q^{-E_{ii}+E_{i+1,i+1}}E_{i,i+1},\quad(q^{E_{ii}})^{*}=q^{E_{ii}}.$

######  Remark 1.

In general, Drinfeld–Jimbo quantum groups have a $*$–Hopf algebra structure
when $q$ is positive. See, for example, sections 9.2.4 and 9.3.5 of [KSBook].
In the context of probability theory, all models will require $q$ to be
positive anyway, so this is not a restrictive requirement.

For any $0\leq i\neq j\leq n$, define the root vectors $E_{ij}$ inductively by

$E_{ij}=E_{ik}E_{kj}-q^{-1}E_{kj}E_{ik},\quad i<k<j\text{ or }i>k>j.$

This definition does not depend on the choice of $k$. With this definition,
there is a central element [GZB91]:

$C_{1}:=\sum_{i=0}^{n}q^{2i-2n-1}q^{2E_{ii}}+\left(q-q^{-1}\right)^{2}\sum_{0\leq
i<j\leq n}q^{2j-2n-2}q^{E_{ii}+E_{jj}}E_{ij}E_{ji}$ (12)

One can check by direction computation that $C_{1}^{*}=C_{1}$. Alternatively,
this can be seen without computations. Namely, the $*$–involution must
preserve the center, and since there is only one first–order Casimir element
(by the Harish–Chandra isomorphism), it must be the case that
$C_{1}^{*}=C_{1}$.

######  Remark 2.

The Lie algebra $\mathfrak{gl}_{n+1}$ is not simple, but contains a simple
sub-algebra $\mathfrak{sl}_{n+1}$ of co–dimension one. The simple Lie algebra
$\mathfrak{sl}_{n+1}$ is the simply the algebra of $(n+1)\times(n+1)$ matrices
with trace zero. The reason for instead using $\mathfrak{gl}_{n+1}$ is that
the central element $C_{1}$ had already been explicitly computed.

Given any $m\geq 1$, let

$\mathcal{B}_{m}^{(n)}=\\{\mu=(\mu_{0},\ldots,\mu_{n}):\mu_{0}+\ldots+\mu_{n}=m\\}\subseteq\mathbb{Z}^{n+1}.$

Let $V^{(n)}_{m}$ be a vector space with basis vectors indexed by
$\mathcal{B}_{m}^{(n)}$. Denote the basis vectors by $V_{\mu}$ for
$\mu\in\mathcal{B}_{m}^{(n)}$. It was proved in Lemma 3.1 of [Kua17] that
$V^{(n)}_{m}$ is a representation of $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$
with action given by

$\displaystyle E_{i,i+1}v_{\mu}$
$\displaystyle=[\mu_{i+1}]_{q}v_{\mu+\epsilon_{i}},\quad 0\leq i<n,$
$\displaystyle E_{i+1,i}v_{\mu}$
$\displaystyle=[\mu_{i}]_{q}v_{\mu-\epsilon_{i}},\quad 0\leq i<n,$
$\displaystyle q^{E_{ii}}v_{\mu}$ $\displaystyle=q^{\mu_{i}}v_{\mu},\quad
0\leq i\leq n,$

where $\epsilon_{i}=(0,0,\ldots,0,1,-1,0,\ldots,0)$ with the $1$ in the $i$th
position. Furthermore, lemma 3.1 of [Kua17] proves that the root vectors
$E_{ij}$ act as

$\displaystyle E_{ij}v_{\mu}$
$\displaystyle=q^{\mu_{i+1}+\mu_{i+2}+\ldots+\mu_{j-1}}\left[\mu_{j}\right]_{q}v_{\mu_{j\rightarrow
i}},$ (13) $\displaystyle E_{ji}v_{\mu}$
$\displaystyle=q^{\mu_{i+1}+\mu_{i+2}+\ldots+\mu_{j-1}}\left[\mu_{i}\right]_{q}v_{\mu_{i\rightarrow
j}},$

where for $i<j$

$\displaystyle\mu_{j\rightarrow i}$
$\displaystyle=\left(\mu_{0},\ldots,\mu_{i-1},\mu_{i}+1,\mu_{i+1},\ldots,\mu_{j-1},\mu_{j}-1,\mu_{j+1},\ldots,\mu_{n}\right),$
(14) $\displaystyle\mu_{i\rightarrow j}$
$\displaystyle=\left(\mu_{0},\ldots,\mu_{i-1},\mu_{i}-1,\mu_{i+1},\ldots,\mu_{j-1},\mu_{j}+1,\mu_{j+1},\ldots,\mu_{n}\right).$

So $\mu_{j\rightarrow i}$ represents the particle configuration where a
species $j$ particle has been replaced with species $i$ particle.

Recall that a $*$–representation $V$ of a $*$–algebra $\mathcal{A}$ is a
representation $V$ that carries an inner product $\langle\cdot,\cdot\rangle$
such that

$\langle av,w\rangle=\langle v,a^{*}w\rangle$

for all $a\in\mathcal{A}$ and $v,w\in V$. It will be shown later that there
exists an inner product such that $V_{m}^{(n)}$ is a $*$–representation.

###  2.4 Markov Duality

######  Definition 2.1.

Two Markov processes $\xi(t)$ and $\eta(t)$ on states spaces $\mathfrak{X}$
and $\mathfrak{Y}$, respectively, are dual with respect to the function
$D(\xi,\eta)$ on $\mathfrak{X}\times\mathfrak{Y}$ if

$\mathbb{E}_{\xi}[D(\xi(t),\eta)]=\mathbb{E}_{\eta}[D(\xi,\eta(t))]\text{ for
all }t\geq 0.$

If, in addition, $\mathfrak{X}=\mathfrak{Y}$ and $\xi(t)=\eta(t)$ then
$\xi(t)$ is self–dual with respect to the function $D(\xi,\eta)$.

If $\mathfrak{X}$ and $\mathfrak{Y}$ are discrete, an equivalent definition of
duality is the intertwining relation

$L_{\xi}^{T}D=DL_{\eta},$

where $L_{\xi}$ is the generator of $\xi(t)$ viewed as a matrix with rows and
columns indexed by $\mathfrak{X}$, $L_{\eta}$ is the generator of $\eta(t)$
with rows and columns indexed by $\mathfrak{Y}$, and $D$ is the duality
function viewed as a matrix with rows indexed by $\mathfrak{X}$ and columns
indexed by $\mathfrak{Y}$. The superscript T indicates matrix transpose. Note
that this is using the convention in mathematical physics, rather than
probability, that a stochastic matrix has columns, rather than rows, summing
to $1$.

Any reversible Markov process is self–dual with respect to the so–called
“cheap” duality function given by the inverses of the reversible measures.
More precisely, if $\xi(t)$ is reversible, then detailed balance reads

$L_{\xi}^{T}=V^{-1}L_{\xi}V,$

where $V$ is the diagonal matrix with entries
$V(\xi,\eta)=\delta_{\xi,\eta}\pi(\xi)$, and $\pi(\xi)$ is a reversible
measure. Then defining $D^{\text{ch}}=V^{-1}$, it is immediate that

$L_{\xi}^{T}D^{\text{ch}}=D^{\text{ch}}L_{\xi}.$

######  Definition 2.2.

If $\xi(t)$ is a Markov process with generator $L_{\xi}$, then a symmetry of
$\xi(t)$ is an operator $S$ such that $L_{\xi}S=SL_{\xi}$.

In the context of self-duality there is a one-to-one correspondence between a
self-duality function and the symmetry of the Markov process. Indeed, if
$\xi(t)$ is a Markov process which is self-dual via $D^{\text{ch}}$ and with
non-trivial symmetry $S$, then the new function
$D^{\text{new}}=SD^{\text{ch}}$ is a different self-duality function. To see
this, note that

$\displaystyle
L_{\xi}^{T}D^{\text{new}}=L_{\xi}^{T}SD^{\text{ch}}=SL_{\xi}^{T}D^{\text{ch}}=SD^{\text{ch}}L_{\xi}=D^{\text{new}}L_{\xi}.$

In this way, applying symmetries to the “cheap” duality function
$D^{\text{ch}}$ produces more interesting, non–trivial self–duality functions.

###  2.5 The $n$–species ASEP$(q,\boldsymbol{\theta})$ and $q$–Hahn Boson

Let $\Lambda_{L}:=\left\\{1,\ldots,L\right\\}$ be a finite one dimensional
chain where all our processes evolves and let
$\boldsymbol{\theta}=(\theta^{x})_{x\in\Lambda_{L}}$ be a family of positive
integers. The $\theta^{x}$ will indicate the maximum number of particles that
may occupy the site $x$. There are $n$ different species of particles, labeled
from $0$ to $n-1$. A particle configuration will be denoted in bold by

$\boldsymbol{\xi}=(\xi^{x}_{i})_{1\leq x\leq L,0\leq i\leq n},$

where $\xi^{x}_{i}$ denotes the number of $i^{th}$ species particles at site
$x$, with $i=n$ denoting a hole. Note that
$\xi^{x}_{0}+\ldots+\xi^{x}_{n}=\theta^{x}$ for all $x\in\Lambda_{L}$. For any
lattice site $x$, The notation $\boldsymbol{\xi}^{x}$ will denote

$\boldsymbol{\xi}^{x}=(\xi^{x}_{0},\ldots,\xi^{x}_{n}),$

which describes the particles located at site $x$ in particle configuration
$\boldsymbol{\xi}$. While $\boldsymbol{\xi}_{i}$ will denote

$\boldsymbol{\xi}_{i}=(\xi^{1}_{i},\ldots,\xi^{L}_{i}),$

which describes the particles of species $i$ in particle configuration
$\boldsymbol{\xi}$.

In terms of jump rates, species with smaller labels are considered to be
heavier or to have priority. The generator is explicitly given by

$\displaystyle\mathcal{L}_{x,x+1}f(\xi)=$ $\displaystyle\sum_{0\leq k<l\leq
n}\left(q^{-1}\cdot
q^{2\left(\xi_{0}^{x}+\ldots+\xi_{k-1}^{x}\right)}\left\\{\xi_{k}^{x}\right\\}_{q^{2}}\cdot
q^{2\left(\xi_{l+1}^{x+1}+\ldots+\xi_{n}^{x+1}\right)}\left\\{\xi_{l}^{x+1}\right\\}_{q^{2}}\left(f\left(\xi_{k\leftrightarrow
l}^{x\leftrightarrow x+1}\right)-f(\xi)\right)\right.$
$\displaystyle\left.+q\cdot
q^{2\left(\xi_{0}^{x}+\ldots+\xi_{l-1}^{x}\right)}\left\\{\xi_{l}^{x}\right\\}_{q^{2}}\cdot
q^{2\left(\xi_{k+1}^{x+1}+\ldots+\xi_{n}^{x+1}\right)}\left\\{\xi_{k}^{x+1}\right\\}_{q^{2}}\left(f\left(\xi_{l\leftrightarrow
k}^{x\leftrightarrow x+1}\right)-f(\xi)\right)\right),$

where $\boldsymbol{\xi}^{x\leftrightarrow x+1}_{i\leftrightarrow j}$ denotes
the particle configuration obtained by switching a species $i$ particle at
lattice site $x$ with a $j^{th}$ species particle at lattice site $x+$1
(assuming that such a particle configuration exists). If this particle
configuration does not exist, then set $\boldsymbol{\xi}^{x\leftrightarrow
x+1}_{i\leftrightarrow j}=\boldsymbol{\xi}$. A verbal description of the model
can be found in [Kua17], which introduced the model. A family of reversible
measures are also given by

$\mu_{\boldsymbol{k}}^{n}(\boldsymbol{\xi})=1_{\left\\{N\left(\xi_{i}\right)=k_{i},0\leq
i\leq
n\right\\}}\prod_{x\in\Lambda_{L}}\prod_{i=0}^{n}\frac{1}{\left[\xi_{i}^{x}\right]_{q}^{!}}q^{\left(\xi_{i}^{x}\right)^{2}/2}\prod_{y<x}\prod_{i=0}^{n-1}q^{-2\xi_{[0,i]}^{x}\xi_{i+1}^{y}},$

where $\xi_{[0,i]}^{x}=\sum_{j=0}^{i}\xi_{j}^{x}$.

When taking the limit where all $\theta^{x}\rightarrow\infty$, the
multi–species ASEP$(q,\boldsymbol{\theta})$ converges to a multi–species
$q$-TAZRP, which was introduced in [Tak], generalizing the single–species
$q$–TAZRP in [SW98]. The jump rates are much simpler to describe, and are
pictorially represented in Figure 2 below.

Figure 2: The jump rates for the multi–species $q$–TAZRP.

Following [KMMO16], define

$\Phi_{q}(\boldsymbol{\gamma}\mid\boldsymbol{\beta};\lambda,\mu)=q^{\chi_{\beta,\gamma}}\left(\frac{\mu}{\lambda}\right)^{|\gamma|}\frac{(\lambda;q)_{|\gamma|}\left(\frac{\mu}{\lambda};q\right)_{|\beta|-|\gamma|}}{(\mu;q)_{|\beta|}}\prod_{i=0}^{n}\binom{\beta_{i}}{\gamma_{i}}_{q},$

where

$\chi_{\beta,\gamma}=\sum_{0\leq i<j\leq
n}\left(\beta_{i}-\gamma_{i}\right)\gamma_{j},\qquad|\gamma|=\sum_{i=0}^{n}\gamma_{i}.$

This defines a discrete–time multi–species $q$–Hahn TAZRP. By taking the
derivative at $\lambda=1$, these can be used to define a continuous–time
multi–species $q$–Hahn TAZRP, which then specializes to the multi–species
$q$–TAZRP. The single–species $q$–Hahn TAZRP, in both discrete and continuous
time, was introduced in [Cor15], using formulas from [Pov13]. The precise
formulas for the generators can be found in section 2.3 of [Kua18a].

###  2.6 Stochastic higher–rank, higher–spin vertex models

The algebraic construction of the duality also holds for the so–called
stochastic higher–rank, higher–spin vertex model. First the model will be
described below.

The stochastic multi–species, higher–rank vertex model is defined from a
stochastic matrix $S(z)$ depending on a spectral parameter $z$; see section
[KMMO16] for definitions and [BoMa16] for an explicit formula. The matrix
$S(z)$ is obtained from the (non–stochastic) $R$–matrix of
$\mathcal{U}_{q}(A_{n}^{(1)})$ by a gauge transform. For the purposes of the
present paper, the most relevant information is that the matrix depends on an
asymmetry parameter $q$, a spectral parameter $z$, and two additional
parameters $\lambda$ and $\mu$. The matrix entries can be visualized as:

$b_{1}$$\begin{subarray}{c}\text{(random)}\\\
\text{output}\end{subarray}$input$1-b_{1}$$\begin{subarray}{c}\text{(random)}\\\
\text{output}\end{subarray}$input Figure 3: The stochastic vertex weights.

The stochastic $S$–matrix defines a discrete–time Markov chain. The state
space for this Markov chain is the same as the state space for the
ASEP$(q,\boldsymbol{\theta})$. As in Figure 4, one fixes boundaries at the
left and bottom, and then updates from left to right. Thus, the arrows on the
bottom are stochastically “updated” to arrows on the top. By viewing this
update as one discrete–time update of a Markov chain, one can construct a
discrete–time Markov chain. In this paper, we impose the boundary conditions
in which no arrows enter from the left or exit from the right. See [Kua18a]
for the mathematically precise definitions.

Figure 4: This figure shows one update of the stochastic vertex model.

##  3 Main Results

With all the notations and processes defined, the main results of the paper
can be stated. The statements of the results are all purely probabilistic,
even though some of the proofs are algebraic.

###  3.1 Orthogonal Polynomial Duality for the multi–species
ASEP$(q,\boldsymbol{\theta})$

The first result regards self-duality for the multi-species
ASEP$(q,\boldsymbol{\theta})$. In order to present it, we first need to recall
and generalize the result for one species only. The function

$\displaystyle
D_{\alpha_{i}}^{\boldsymbol{\theta}}\left(\boldsymbol{\xi}_{i},\boldsymbol{\eta}_{i}\right)=\prod_{x=1}^{L}K_{\boldsymbol{\eta}_{i}^{x}}\left(q^{-2\xi_{i}^{x}},p_{i}^{x}\left(\boldsymbol{\xi}_{i},\boldsymbol{\eta}_{i}\right),\theta^{x},q^{2}\right)$
(15) $\displaystyle\text{with}\
p_{i}^{x}\left(\boldsymbol{\xi}_{i},\boldsymbol{\eta}_{i}\right)=\alpha_{i}^{-1}q^{-2\left(N_{x-1}^{-}\left(\boldsymbol{\xi}_{i}\right)-N_{x+1}^{+}\left(\boldsymbol{\eta}_{i}\right)\right)+2N_{x-1}^{-}(\boldsymbol{\theta})-1}$

generalizes the single species homogeneous case to be inhomogeneous. Above
$\displaystyle{N_{x-1}^{-}\left(\boldsymbol{\xi}_{i}\right)=\sum_{1\leq y\leq
x-1}\xi_{i}^{y}}$ and
$\displaystyle{N_{x+1}^{+}\left(\boldsymbol{\eta}_{i}\right)=\sum_{x+1\leq
y\leq L}\eta_{i}^{y}}$ denotes the number of particles in the configuration
considered at the left, respectively right, of site $x$. Indeed, this
$D_{\alpha}^{\boldsymbol{\theta}}$ is up to a constant the orthogonal
self–duality function for ASEP$(q,\theta)$, which was proven in Theorem 3.2 of
[CFG21]. Below, in the section of proofs, we will see that this is the
orthogonal self–duality function in the inhomogeneous case as well.

######  Remark 3.

The choice of introducing a maximal occupancy $\theta^{x}\in\mathbb{N}$ to
each site $x\in\Lambda_{L}$ can be interpreted as a random environment for the
exclusion process. Here the maximal occupancy is not fixed and can vary among
sites as the microscopic dynamics occur. Interacting particle systems in
random environment have been considered extensively in the context of scaling
limit where one is interested in the macroscopic effects of the environment.
From a duality outlook, we mention the following result [FRS], where the
authors extend the orthogonal polynomial dualities for symmetric particle
systems in a general graph where each vertex allows an arbitrary number of
particles with an open boundary.

Now proceed to the $n$–species case. Given two particle configurations
$\boldsymbol{\xi}$ and $\boldsymbol{\eta}$ of the $n$–species
ASEP$(q,\boldsymbol{\theta})$ on $L$ lattice sites, define intermediate
particle configurations $\boldsymbol{\zeta}^{(i)}$ by

$\displaystyle\boldsymbol{\eta}$
$\displaystyle=(\boldsymbol{\eta}_{0},\boldsymbol{\eta}_{1},\ldots,\boldsymbol{\eta}_{n-1},\boldsymbol{\eta}_{n}),$
$\displaystyle\boldsymbol{\zeta}^{(1)}$
$\displaystyle=(\boldsymbol{\xi}_{0},\boldsymbol{\eta}_{1}+\boldsymbol{\eta}_{0}-\boldsymbol{\xi}_{0},\ldots,\boldsymbol{\eta}_{n-1},\boldsymbol{\eta}_{n}),$
$\displaystyle\ldots$ $\displaystyle=\ldots,$
$\displaystyle\boldsymbol{\zeta}^{(n-1)}$
$\displaystyle=(\boldsymbol{\xi}_{0},\boldsymbol{\xi}_{1},\ldots,\boldsymbol{\xi}_{n-1}+\boldsymbol{\xi}_{n}-\boldsymbol{\eta}_{n},\boldsymbol{\eta}_{n}),$
$\displaystyle\boldsymbol{\xi}$
$\displaystyle=(\boldsymbol{\xi}_{0},\boldsymbol{\xi}_{1}\ldots,\boldsymbol{\xi}_{n-1},\boldsymbol{\xi}_{n}).$

More precisely, $\boldsymbol{\zeta}^{(i)}$ is the unique particle
configuration of $n$–species ASEP$(q,\boldsymbol{\theta})$ on $L$ lattice
sites with the property that

$\boldsymbol{\zeta}^{(i)}_{k}=\begin{cases}\boldsymbol{\xi}_{k},\quad\text{ if
}k<i\\\ \boldsymbol{\eta}_{k},\quad\text{ if }k>i.\end{cases}$

The value of $\boldsymbol{\zeta}^{(i)}_{i}$ is uniquely determined by
$\sum_{k=0}^{n}\boldsymbol{\zeta}^{(i)}_{k}=\boldsymbol{\theta}$, i.e.
$\boldsymbol{\zeta}^{(i)}_{i}=\boldsymbol{\eta}_{[0,i]}-\boldsymbol{\xi}_{[0,i-1]}$,
where $\boldsymbol{\eta}_{[0,i]}$ is the vector
$\sum_{j=0}^{i}\boldsymbol{\eta}_{j}$. Additionally, define intermediate
values of $\boldsymbol{\theta}^{(i)}$ by

$\boldsymbol{\theta}^{(0)}=\boldsymbol{\eta}_{0}+\boldsymbol{\eta}_{1},\boldsymbol{\theta}^{(1)}=\boldsymbol{\eta}_{1}+\boldsymbol{\eta}_{2}-\boldsymbol{\xi}_{0}+\boldsymbol{\eta}_{0},\quad\ldots,\quad\boldsymbol{\theta}^{(n-2)}=\boldsymbol{\xi}_{n-2}+\boldsymbol{\xi}_{n-1}-\boldsymbol{\eta}_{n-1}+\boldsymbol{\xi}_{n-3},\quad\boldsymbol{\theta}^{(n-1)}=\boldsymbol{\xi}_{n-1}+\boldsymbol{\xi}_{n}.$

Equivalently,

$\boldsymbol{\theta}^{(i)}=\boldsymbol{\zeta}^{(i)}_{i}+\boldsymbol{\zeta}^{(i)}_{i+1}=\boldsymbol{\eta}_{[0,i+1]}-\boldsymbol{\xi}_{[0,i-1]}.$

For the multi–species case, define

$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})=\mathcal{G}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})\prod_{i=0}^{n-1}D^{\boldsymbol{\theta}^{(i)}}_{\alpha_{i}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i}),$

where

$\mathcal{G}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})=\sqrt{\frac{\prod_{i=0}^{n-1}\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\boldsymbol{\eta})}}.$

Also,

$\mu_{\alpha}^{\boldsymbol{\theta}}(\boldsymbol{\xi}):=\prod_{x=1}^{L}\alpha^{\xi^{x}}\left(\begin{array}[]{l}\theta^{x}\\\
\xi^{x}\end{array}\right)_{q}q^{-\left(2N_{x-1}^{-}(\boldsymbol{\theta})+\theta^{x}\right)\xi^{x}},$

is a reversible measure for the single–species ASEP$(q,\boldsymbol{\theta})$,
and

$\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta}):=\sqrt{\prod_{i=0}^{n-1}q^{\binom{N(\boldsymbol{\xi}_{i})}{2}-\binom{N(\boldsymbol{\zeta}^{(i)}_{i})}{2}}\frac{\left(\alpha_{i}q^{1-N(\boldsymbol{\eta}_{[0,i+1]}-\boldsymbol{\xi}_{[0,i]})}\right)_{\infty}}{\left(\alpha_{i}q^{1-N(\boldsymbol{\eta}_{[0,i]}-\boldsymbol{\xi}_{[0,i-1]})}\right)_{\infty}}}$

Note that
$\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})$ is
constant with respect to the dynamics, but is needed for the orthogonality
below.

The first main theorem is the following and it states that
$\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})$
is an orthogonal self-duality function for the multi-species exclusion
process.

######  Theorem 3.1.

(a) The multi–species ASEP$(q,\boldsymbol{\theta})$ is self–dual with respect
to the function
$\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})$.

(b) Let $(a_{\boldsymbol{k}})$ be a family of non–negative (positive?) real
numbers which sum to $1$. Then

$\mu^{n}(\boldsymbol{\eta}):=\sum_{\boldsymbol{k}}a_{\boldsymbol{k}}\mu^{n}_{\boldsymbol{k}}(\boldsymbol{\eta})$

is a reversible measure for the multi–species ASEP($q,\boldsymbol{\theta}$),
and furthermore the duality functions from part (a) are orthogonal with
respect to these reversible measures in the sense that

$\sum_{\boldsymbol{\xi}}\mu^{n}(\boldsymbol{\xi})\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\bar{\boldsymbol{\eta}})\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\bar{\boldsymbol{\eta}})=\begin{cases}\frac{1}{\mu^{n}(\boldsymbol{\eta})},\quad\boldsymbol{\eta}=\bar{\boldsymbol{\eta}},\\\
0,\quad\boldsymbol{\eta}\neq\bar{\boldsymbol{\eta}}.\end{cases}$

This theorem generalizes a few previously known results.

  * •

In the single–species case, when $n=1$, the results of [CFG21] are recovered,
namely an inhomomgeneous nested product of $q$-Krawtchouk polynomials.

  * •

In the symmetric case, when $q\xrightarrow[]{}1$, the duality function reduces
to a special case of the homogeneous product of multivariate Krawtchouk
polynomials [Zho], using the definition of multivariate Krawtchouk polynomials
from Appendix 2 in [Mil68].

  * •

If the two conditions above are considered, i.e. setting $n=1$ and taking the
$q\xrightarrow[]{}1$ limit, one gets an homogeneous product of Krawtchouk
polynomials as self-duality function, see for example [Gro19].

###  3.2 Example

Before proceeding with the next results we look at three different concrete
examples where the maximum number of particles for each sites is fixed and
equal to $2$.

######  Example 1.

Suppose $n=2$, then the self–duality function is

$\begin{array}[]{c}\mathcal{D}(\boldsymbol{\xi},\boldsymbol{\eta})=D_{\alpha_{0}}^{\left(\boldsymbol{\eta}_{0}+\eta_{1}\right)}\left(\boldsymbol{\xi}_{0},\boldsymbol{\eta}_{0}\right)D_{\alpha_{1}}^{\left(\boldsymbol{\theta}-\boldsymbol{\xi}_{1}\right)}\left(\boldsymbol{\xi}_{1},\boldsymbol{\eta}_{0}+\boldsymbol{\eta}_{1}-\boldsymbol{\xi}_{1}\right)\sqrt{\frac{\mu^{(\boldsymbol{\eta}_{0}+\boldsymbol{\eta}_{1})}_{\alpha_{0}}(\boldsymbol{\xi}_{0})\mu^{(\boldsymbol{\eta}_{0}+\boldsymbol{\eta}_{1})}_{\alpha_{0}}(\boldsymbol{\eta}_{0})\mu^{(\boldsymbol{\theta}-\boldsymbol{\xi}_{0})}_{\alpha_{1}}(\boldsymbol{\xi}_{1})\mu^{(\boldsymbol{\theta}-\boldsymbol{\xi}_{0})}_{\alpha_{1}}(\boldsymbol{\eta}_{0}+\boldsymbol{\eta}_{1}-\boldsymbol{\xi}_{0})}{\mu^{3}(\boldsymbol{\xi})\mu^{3}(\boldsymbol{\eta})}}.\end{array}$

For $\theta^{x}\equiv 2,L=2$ consider the four particle configurations

$\begin{array}[]{llllllllllll}0&2&&2&2&&2&2&&2&0\\\
\underline{1}&\underline{2}&&\underline{1}&\underline{0}&&\underline{0}&\underline{1}&&\underline{2}&\underline{1}\end{array}$

where $2$ denotes a hole. Using these four particle configurations as an
ordered basis, the duality function is expressed as the matrix

$\mathcal{D}=\begin{bmatrix}(q^{2}+1)(\alpha_{1}-q)C_{0}/q^{8}&\alpha_{1}(q^{2}+1)(\alpha_{0}-q)/q^{6}&\alpha_{0}\alpha_{1}(q+1/q)/q^{5}&0\\\
\alpha_{1}(q^{2}+1)C_{0}/q^{8}&(\alpha_{0}-q)(-q^{5}+\alpha_{1}q^{2}+\alpha_{1})/q^{6}&\alpha_{0}(-q^{5}+\alpha_{1}q^{2}+\alpha_{1})/q^{6}&0\\\
0&\alpha_{0}(-q^{3}+\alpha_{1}q^{2}+\alpha_{1})/q^{4}&(-q^{3}+\alpha_{0})*(-q^{3}+\alpha_{1}q^{2}+\alpha_{1})/q^{4}&\alpha_{1}(q^{2}+1)*C_{0}/q^{4}\\\
0&\alpha_{0}\alpha_{1}(q+1/q)/q^{3}&\alpha_{1}(q^{2}+1)*(-q^{3}+\alpha_{0})/q^{4}&(q^{2}+1)*(-q^{5}+\alpha_{1})C_{0}/q^{4}\par\end{bmatrix}$
(16)

where $C_{0}$ is the constant $(-q^{3}+\alpha_{0}q^{2}+\alpha_{0})$. With
respect to the same basis, the generator is

$\mathcal{L}=\left(\begin{array}[]{cccc}-q^{3}\left(q+q^{-1}\right)^{2}&q^{3}\left(1+q^{2}\right)&q\left(1+q^{2}\right)&0\\\
q^{7}&-\left(q+q^{3}+q^{7}\right)&q^{3}&q\\\
q^{7}&q^{5}&-\left(q+q^{5}+q^{7}\right)&q\\\
0&q^{5}\left(1+q^{2}\right)&q^{3}\left(1+q^{2}\right)&-q^{5}\left(q+q^{-1}\right)^{2}\end{array}\right)$

######  Example 2.

Suppose $L=3,n=2$, $\theta^{x}\equiv 2$, Let $X_{t},Y_{t}$ be two multi-
species ASEP with initial condition $x_{1},y_{1}$ as following.

$x_{1}=\begin{matrix}2&2&2\\\
\underline{0}&\underline{2}&\underline{2}\end{matrix}\ \ \ \
y_{1}=\begin{matrix}2&2&2\\\
\underline{1}&\underline{2}&\underline{2}\end{matrix}$

$x_{2}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{0}&\underline{2}\end{matrix}\ \ \ \
y_{2}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{1}&\underline{2}\end{matrix}\ \ \ \
x_{3}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{2}&\underline{0}\end{matrix}\ \ \ \
y_{3}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{2}&\underline{1}\end{matrix}$

Then $X_{t},Y_{t}$ reduce to single particle ASEP with the same jumping rates.

$\mathbb{E}_{x_{1}}[D(X_{t},y_{1})]=\sum_{i=0}^{2}\mathbb{P}_{x_{1}}(X_{t}=x_{i})D(x_{i},y_{1}).$

Note that when $i\neq 1$, $D(x_{i},y_{1})=D(x_{1},y_{i})=0$, thus the duality
relation holds.

######  Example 3.

Suppose $L=3,n=2$, $\theta^{x}\equiv 2$, Let $X_{t},Y_{t}$ be two multi-
species ASEP with initial condition $x_{1},y_{1}$ as following. Then the
evolution of species $1$ particle in both process is the same.

$x_{1}=\begin{matrix}2&2&2\\\
\underline{0}&\underline{2}&\underline{2}\end{matrix}\ \ \ \
y_{1}=\begin{matrix}1&1&1\\\
\underline{0}&\underline{1}&\underline{1}\end{matrix}$

$x_{2}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{0}&\underline{2}\end{matrix}\ \ \ \
y_{2}=\begin{matrix}1&1&1\\\
\underline{1}&\underline{0}&\underline{1}\end{matrix}\ \ \ \
x_{3}=\begin{matrix}2&2&2\\\
\underline{2}&\underline{2}&\underline{0}\end{matrix}\ \ \ \
y_{3}=\begin{matrix}1&1&1\\\
\underline{1}&\underline{1}&\underline{0}\end{matrix}$

One can check that $D(x_{i},y_{1})=D(x_{1},y_{i})$ for any $i$. By the fact
that $\mathbb{P}_{x_{1}}(X_{t}=x_{i})=\mathbb{P}_{y_{1}}(Y_{t}=y_{i})$, the
duality still holds.

###  3.3 Duality for the multi–species $q$–TAZRP

The theorem is a duality result for the multi–species totally asymmetric zero
range process. The so–called charge–parity transformation $T$ was introduced
in [Kua17] in the multi–species case. It is an involution which satisfies the
property that $TLT^{-1}=L^{\text{rev}}$, where $L$ is the generator of
multi–species ASEP$(q,\boldsymbol{\theta})$ and $L^{\text{rev}}$ is the
space–reversed generator..
The idea is that on site $x\in\Lambda_{L}$ we replace the $i^{th}$ species
with the $(n-i)^{th}$ one for $0\leq i\leq n$, namely
$(T(\boldsymbol{\xi}))^{x}_{i}:=\xi^{x}_{n-i}.$ Once the charge–parity
transformation is applied and the $\boldsymbol{\theta}\rightarrow\infty$ limit
is considered, the duality function
$\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})$
yields a new duality function for general $n$-species $q$-TAZRP ($n\geq 2$).

By applying a certain transformation, called the charge–parity transformation,
the $\boldsymbol{\theta}\rightarrow\infty$ limit of the duality function
yields a duality for general $n$-species $q$-TAZRP ($n\geq 2$).

######  Theorem 3.2.

Let $T$ be the involution $(T(\boldsymbol{\xi}))^{x}_{i}=\xi^{x}_{n-i}$ for
$1\leq x\leq L,0\leq i\leq n$. Then the function

$\displaystyle
D^{q-TARZP}(\boldsymbol{\xi},\boldsymbol{\eta})=q^{h(\boldsymbol{\xi},\boldsymbol{\eta})}\prod_{i=0}^{n-1}\prod_{x\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-2\xi_{i}^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1}\right)$
(17)
$\displaystyle=q^{h(\boldsymbol{\xi},\boldsymbol{\eta})}\prod_{i=0}^{n-1}\prod_{x\in\Lambda_{L}}(q^{-2\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1};q^{2})_{2\xi_{i}^{x}}$

is a space reversed duality for $n$-species $q$-TAZRP ($n\geq 1$), where
$\boldsymbol{\eta}$ evolves with total asymmetry to the left and
$\boldsymbol{\xi}$ evolves with total asymmetry to the right, and

$h(\boldsymbol{\xi},\boldsymbol{\eta})=\sum_{x\in\Lambda_{L}}\sum_{i=0}^{n-1}\left(-\xi_{i}^{x}N_{x+1}^{+}(\boldsymbol{\eta}_{[0,n-2-i]})+\eta_{i}^{x}N_{x}^{+}(\boldsymbol{\xi}_{[0,n-2-i]})\right).$
(18)

Note that when $\boldsymbol{\eta}$ consists of finitely many sites each
occupied with infinitely many particles, and $\boldsymbol{\xi}$ consists of
finitely many particles, the duality function becomes an indicator function.

###  3.4 Stochastic higher–spin, higher–rank vertex models

In [Kua18a], it was proved that after applying charge–parity symmetry of the
previous section, the triangular duality function for the multi–species
ASEP$(q,\boldsymbol{\theta})$ is also a duality function for the stochastic
higher–spin, higher–rank vertex models. The argument used there immediately
implies the same for the orthogonal polynomial duality function, as stated
below.

######  Theorem 3.3.

The stochastic higher–spin, higher–rank vertex model, with total asymmetry to
the right, is dual to its space reversal with respect to the function

$\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},T(\boldsymbol{\eta})).$

Note that taking the $\boldsymbol{\theta}\rightarrow\infty$ limit recovers
Theorem 3.2.

###  3.5 Orthogonal Polynomial Duality for the multi–species $q$–Hahn TAZRP

In [Kua18a], it was shown that the triangular duality for the multi–species
$q$–Boson was also a duality for the multi–species $q$–Hahn Boson. The proof
proceeded by direct computation. As such, it is natural to check if the
orthogonal duality for the multi–species $q$–Boson is also a duality for the
multi–species $q$–Hahn Boson. Indeed, this turns out to be the case:

######  Theorem 3.4.

The multi–species $q$–Hahn TAZRP (with drift to the right) is dual to its
space reversal (with drift to the left), with respect to the same function as
in Theorem 3.2.

###  3.6 Application of the Duality

We end this section with an application of the duality function for the
two–species $q$-TARZP in equation (17). It is natural to exploit the duality
relation for the computation of suitable moments of the process. In
particular, one expects interacting particle systems to exhibit long range
correlations arising as a consequence of the dual microscopic interaction.
Using a simple initial configuration, the duality function can give
information on such quantities, which are explicitly known only for few
special cases. However, in the context of multi-species asymmetric particle
systems, applications of duality are still under investigation. In [BS18] the
authors took advantage of the self-duality of multi-species exclusion process
to obtain all invariant measures and to study the microscopic shocks dynamics.
Below we show that, choosing a dual configuration $\boldsymbol{\eta}$
consisting of only two particles of different species and following their
dynamics, we get a close formula for the expectation with respect to the
initial process $\boldsymbol{\xi}$ in terms of the product of two $q-$shifted
factorials.

Recall the incomplete gamma function

$\Gamma(a,z)=\int_{z}^{\infty}t^{a-1}\mathrm{e}^{-t}\mathrm{~{}d}t$

and define its normalization

$Q(a,z)=\frac{\Gamma(a,z)}{\Gamma(a)}.$

######  Theorem 3.5.

Suppose $\boldsymbol{\xi}$ has initial conditions with $n_{1}$ species 0
particles at site $x_{1}$ and $n_{2}$ species 1 particles at site $x_{2}$, or
in other words

$\boldsymbol{\xi}_{i}^{x}=\begin{cases}n_{1}\ \text{for}\ i=0,x=x_{1},\\\
n_{2}\ \text{for}\ i=1,x=x_{2},\\\ 0\ \ \text{else},\end{cases}$

and $\boldsymbol{\xi}$ evolves as a $q$–TAZRP with total asymmetry to the
right. Let $y_{1}$ and $y_{2}$ be some lattice sites, and additionally
assuming that $x_{1}<x_{2}$.

The dual process $\boldsymbol{\eta}$ has 1 species 0 particles at site
$y_{1}$, 1 species 1 particles at site $y_{2}$, so in other words

$\boldsymbol{\eta}_{i}^{y}=\begin{cases}1\ \text{for}\ i=0,y=y_{1},\\\ 1\
\text{for}\ i=1,y=y_{2},\\\ 0\ \ \text{else}.\end{cases}$

Then there is the explicit formula

$\mathbb{E}_{\boldsymbol{\xi}}[q^{h(\boldsymbol{\xi}(t),\boldsymbol{\eta})}\prod_{i=0}^{1}\prod_{x=1\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-2\xi_{i}^{x}(t)};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i}(t))+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1}\right)]=\sum_{i=1}^{6}{\delta_{i}p_{i}},$

where

$\displaystyle\delta_{1}:=q^{n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}},$
$\displaystyle\delta_{2}:=q^{n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}},$
$\displaystyle\delta_{3}:=q^{-n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}},$
$\displaystyle\delta_{4}:=q^{-n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}},$
$\displaystyle\delta_{5}:=q^{-n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}},$
$\displaystyle\delta_{6}:=q^{-n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}},$

and

$p_{1}=q_{1}-q_{4},\quad p_{2}=q_{4},\quad p_{3}=q_{2}+q_{3}-q_{5}-q_{6},\quad
p_{4}=-q_{3}+q_{5}+q_{6},\quad p_{5}=q_{5},\quad p_{6}=q_{3}-q_{5}.$

with

$q_{1}=Q(y_{1}-x_{1},t),\quad q_{3}=1-Q(y_{2}-x_{2},t),\quad
q_{2}=1-q_{2}-q_{3}$

and

$\displaystyle q_{4}$ $\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{C}\frac{dw_{1}}{w_{1}}\int_{C}\frac{dw_{2}}{w_{2}}\frac{w_{1}-w_{2}}{w_{1}-qw_{2}}\prod_{j=1}^{2}\left[(1-w_{j})^{-(x_{1}-y_{j}+1)}e^{-w_{j}t}\right]$
$\displaystyle q_{6}$ $\displaystyle=1-q_{1}-\frac{q}{(2\pi
i)^{2}}\int_{\tilde{C}_{1}}\frac{dw_{1}}{w_{1}}\int_{\tilde{C}_{2}}\frac{dw_{2}}{w_{2}}\frac{w_{1}-w_{2}}{w_{1}-qw_{2}}\prod_{j=1}^{2}\left[(1-w_{j})^{-(x_{1}-y_{j}+1)}e^{-w_{j}t}\right]$
$\displaystyle q_{5}$ $\displaystyle=\frac{1}{(2\pi
i)^{2}}\int_{\tilde{C}_{1}}\frac{dw_{1}}{w_{1}}\int_{\tilde{C}_{2}}\frac{dw_{2}}{w_{2}}\frac{w_{1}-w_{2}}{w_{1}-qw_{2}}\prod_{j=1}^{2}(1-w_{j})^{-(x_{1}-y_{j}+1)}e^{-w_{j}t}.$

######  Remark 4.

An asymptotic analysis of the two–point correlations can be done, but requires
technical details which are better-suited for a future paper.

######  Remark 5.

Note that in the $x_{2}\geq x_{1}$ situation, if furthermore
$y_{1},y_{2}>x_{2}$, then the expectation of the moments trivially reduces to
a single–species $q$–TAZRP when there are infinitely many first–class
particles. This is because the jump rate for a second–class particle is
multiplied by $q^{\infty}=0$, so a jump for a second–class particle never
occurs once it reaches $x_{1}$.

##  4 Proofs

We devote this section to the rigorous proof of the previous results.

######  Remark 6.

The proofs presented in the next section generalize to other quantum groups
and representations. In an upcoming paper by the second and third authors, the
method is applied to the quantum group $\mathcal{U}_{q}(\mathfrak{so}_{6})$
and the type $D$ ASEP [BBKLUZ].

###  4.1 A direct proof of Theorem 3.1(b)

The orthogonality result in Theorem 3.1(b) can be proved directly using
special functions, without prior algebraic and probability background. As
such, it will be proved first. Before doing so, here is a lemma which
generalizes Proposition 7.1 of [CFG21] for the inhomogeneous case of the
single–species ASEP$(q,\boldsymbol{\theta})$.

Given two single-species particle configurations $\boldsymbol{\xi}$ and
$\boldsymbol{\eta}$, define

$K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})=\prod_{x}K_{\eta^{x}}\left(q^{-\xi^{x}},pq^{N_{x-1}^{-}(\boldsymbol{\theta})}q^{-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})},\theta^{x},q\right)$
(19)

and also let

$w_{\boldsymbol{\theta}}(\boldsymbol{\xi})=\frac{(-1)^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})}}{\left(pq^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})+1}\right)_{\infty}}\prod_{x=1}^{L}\frac{q^{\binom{\xi^{x}}{2}}q^{\theta^{x}N_{x-1}^{-}(\boldsymbol{\xi})}}{(q)_{\xi^{x}}(q)_{\theta^{x}-\xi^{x}}},$
(20)
$h_{\boldsymbol{\theta}}(\boldsymbol{\eta})=\frac{(-1)^{N(\boldsymbol{\eta})}q^{\binom{N(\boldsymbol{\theta})+1}{2}}p^{N(\boldsymbol{\theta})}}{\prod_{x}[(q)_{\theta^{x}}]^{2}\left(pq^{N(\boldsymbol{\eta})+1}\right)_{\infty}}\prod_{x=1}^{L}\frac{(q)_{\eta^{x}}(q)_{\theta^{x}-\eta^{x}}}{q^{\binom{\eta^{x}+1}{2}}}q^{\theta^{x}N_{x}^{+}(\boldsymbol{\eta})}.$
(21)

######  Lemma 4.1.

There is the orthogonality relation

$\sum_{\boldsymbol{\xi}}\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\bar{\eta}})=\delta_{\boldsymbol{\eta},\bar{\boldsymbol{\eta}}}.$

###### Proof.

We start with the standard orthogonality relation for $q$-Krawtchouk (equation
(112) of [CFG] or section 1.14.14 of [KLS])

$\sum_{k=0}^{\theta}\frac{(pq)_{\theta-k}(-1)^{\theta-k}q^{\binom{k}{2}}}{(q)_{k}(q)_{\theta-k}}K_{m}(q^{-k},p,\theta,q)K_{n}(q^{-k},p,\theta,q)=\\\
\delta_{m,n}\frac{(-1)^{n}p^{\theta}(q)_{\theta-n}(q)_{n}(pq)_{n}}{[(q)_{\theta}]^{2}}q^{\binom{\theta+1}{2}-\binom{n+1}{2}+\theta
n}.$ (22)

Divide both sides by $(pq)_{\infty}$ and use (1.8.8) of [KLS] to get

$\sum_{k=0}^{\theta}\frac{(-1)^{\theta-k}q^{\binom{k}{2}}}{(q)_{k}(q)_{\theta-k}(pq^{1+\theta-k})_{\infty}}K_{m}(q^{-k},p,\theta,q)K_{n}(q^{-k},p,\theta,q)=\\\
\delta_{m,n}\frac{(-1)^{n}p^{\theta}(q)_{\theta-n}(q)_{n}}{[(q)_{\theta}]^{2}(pq^{1+n})_{\infty}}q^{\binom{\theta+1}{2}-\binom{n+1}{2}+\theta
n}.$ (23)

Define now

$W(k,p,\theta,q)=\frac{(-1)^{\theta-k}q^{\binom{k}{2}}}{(q)_{k}(q)_{\theta-k}(pq^{1+\theta-k})_{\infty}},$
(24)
$H(n,p,\theta,q)=\frac{(-1)^{n}p^{\theta}(q)_{\theta-n}(q)_{n}}{[(q)_{\theta}]^{2}(pq^{1+n})_{\infty}}q^{\binom{\theta+1}{2}-\binom{n+1}{2}+\theta
n}.$ (25)

Let

$K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})=\prod_{x=1}^{L}K_{\eta^{x}}\left(q^{-\xi^{x}},pq^{N_{x-1}^{-}(\boldsymbol{\theta})}q^{-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})},\theta^{x},q\right).$
(26)

and moreover define

$W^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})=W(\xi^{x},pq^{N^{-}_{x-1}(\boldsymbol{\theta})}q^{-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})},\theta^{x},q),$
(27)
$H^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})=H(\eta^{x},pq^{N^{-}_{x-1}(\boldsymbol{\theta})}q^{-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})},\theta^{x},q),$
(28)
$K^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})=K_{\eta^{x}}(\xi^{x},pq^{N^{-}_{x-1}(\boldsymbol{\theta})}q^{-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})},\theta^{x},q).$
(29)

Next we show that
$\prod_{x=1}^{L}\frac{W_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}{H_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}$
can be written as
$\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}$.

$\begin{array}[]{l}\prod_{x=1}^{L}\frac{W^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}{H^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}\\\
=\prod_{x=1}^{L}\frac{(-1)^{\theta^{x}-\xi^{x}-\eta^{x}}q^{\binom{\xi^{x}}{2}}q^{\binom{\eta^{x}+1}{2}-\theta^{x}\eta^{x}}\left(pq^{N^{-}_{x-1}(\boldsymbol{\theta})-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})+\eta^{x}+1}\right)_{\infty}[(q)_{\theta^{x}}]^{2}q^{\theta^{x}N_{x-1}^{-}(\boldsymbol{\xi})}}{(q)_{\xi^{x}}(q)_{\theta^{x}-\xi^{x}}(q)_{\eta^{x}}(q)_{\theta^{x}-\eta^{x}}q^{\binom{\theta^{x}+1}{2}}\left(pq^{N^{-}_{x-1}(\boldsymbol{\theta})-N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})+\theta^{x}-\xi^{x}+1}\right)_{\infty}p^{\theta^{x}}q^{\theta^{x}N^{-}_{x-1}(\boldsymbol{\theta})}q^{\theta^{x}N_{x+1}^{+}(\boldsymbol{\eta})}}\\\
=\frac{(-1)^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})}\left(pq^{N(\boldsymbol{\eta})+1}\right)_{\infty}\prod_{x}[(q)_{\theta^{x}}]^{2}}{(-1)^{N(\boldsymbol{\eta})}q^{\binom{N(\boldsymbol{\theta})+1}{2}}p^{N(\boldsymbol{\theta})}\left(pq^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})+1}\right)_{\infty}}\prod_{x=1}^{L}\frac{q^{\binom{\xi^{x}}{2}}q^{\binom{\eta^{x}+1}{2}}q^{\theta^{x}N_{x-1}^{-}(\boldsymbol{\xi})}}{(q)_{\xi^{x}}(q)_{\theta^{x}-\xi^{x}}(q)_{\eta^{x}}(q)_{\theta^{x}-\eta^{x}}q^{\theta^{x}N_{x}^{+}(\boldsymbol{\eta})}}.\end{array}$
(30)

So now we let

$w_{\boldsymbol{\theta}}(\boldsymbol{\xi})=\frac{(-1)^{N(\theta)-N(x)}}{\left(pq^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})+1}\right)_{\infty}}\prod_{x=1}^{L}\frac{q^{\binom{\xi^{x}}{2}}q^{\theta^{x}N_{x-1}^{-}(\xi)}}{(q)_{\xi^{x}}(q)_{\theta^{x}-\xi^{x}}},$
(31)
$h_{\boldsymbol{\theta}}(\boldsymbol{\eta})=\frac{(-1)^{N(\boldsymbol{\eta})}q^{\binom{N(\boldsymbol{\theta})+1}{2}}p^{N(\boldsymbol{\theta})}}{\prod_{x}[(q)_{\theta^{x}}]^{2}\left(pq^{N(\boldsymbol{\eta})+1}\right)_{\infty}}\prod_{x=1}^{L}\frac{(q)_{\eta^{x}}(q)_{\theta^{x}-\eta^{x}}}{q^{\binom{\eta^{x}+1}{2}}}q^{\theta^{x}N_{x}^{+}(\boldsymbol{\eta})}.$
(32)

Thus

$\sum_{\boldsymbol{\xi}}\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\bar{\eta}})=\prod_{x=1}^{L}\sum_{\xi^{x}=0}^{\theta^{x}}\frac{W^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}{H^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}K^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})K^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\bar{\boldsymbol{\eta}}).$

By the orthogonal relation

$\sum_{\xi^{x}=0}^{\theta^{x}}\frac{W^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}{H^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})}K^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\boldsymbol{\eta})K^{\boldsymbol{\theta}}_{x}(\boldsymbol{\xi},\bar{\boldsymbol{\eta}})=\delta_{\eta^{x},\bar{\eta}^{x}},$
(33)

we get

$\sum_{\boldsymbol{\xi}}\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}K^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})K^{\boldsymbol{\theta}}(\boldsymbol{\boldsymbol{\xi},\bar{\eta}})=\delta_{\boldsymbol{\eta},\bar{\boldsymbol{\eta}}},$

or in other words,
$K^{\boldsymbol{\theta}}(\boldsymbol{\eta},\boldsymbol{\xi})$ and
$K^{\boldsymbol{\theta}}(\boldsymbol{\bar{\eta}},\boldsymbol{\xi})$ are
orthogonal with respect to $w_{\boldsymbol{\theta}}(\boldsymbol{\xi})$ with
norm $h_{\boldsymbol{\theta}}(\boldsymbol{\eta})$.

In addition,to recover the orthogonal relation appeared in [CFG21], we can
rewrite
$\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}$
as the following

$\begin{array}[]{cl}\frac{w_{\boldsymbol{\theta}}(\boldsymbol{\xi})}{h_{\boldsymbol{\theta}}(\boldsymbol{\eta})}&=\frac{(-1)^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})}\left(pq^{N(\boldsymbol{\eta})+1}\right)_{\infty}\prod_{x}[(q)_{\theta^{x}}]^{2}}{(-1)^{N(\boldsymbol{\eta})}q^{\binom{N(\boldsymbol{\theta})+1}{2}}p^{N(\boldsymbol{\theta})}\left(pq^{N(\boldsymbol{\theta})-N(\boldsymbol{\xi})+1}\right)_{\infty}}\displaystyle\prod_{x=1}^{L}\frac{q^{\binom{\xi^{x}}{2}}q^{\binom{\eta^{x}+1}{2}}q^{\theta^{x}N_{x-1}^{-}(\boldsymbol{\xi})}}{(q)_{\xi^{x}}(q)_{\theta^{x}-\xi^{x}}(q)_{\eta^{x}}(q)_{\theta^{x}-\eta^{x}}q^{\theta^{x}N_{x}^{+}(\boldsymbol{\eta})}}\\\
&=\frac{(p^{-1}q^{-N(\boldsymbol{\theta}-\boldsymbol{\xi})})_{\infty}q^{\frac{N(\boldsymbol{\xi})^{2}}{2}-\frac{N(\boldsymbol{\eta})^{2}}{2}}}{(p^{-1}q^{-N(\boldsymbol{\eta})})_{\infty}p^{N(\boldsymbol{\xi})+N(\boldsymbol{\eta})}}\displaystyle{\prod_{x}}\binom{\theta^{x}}{\xi^{x}}_{q^{\frac{1}{2}}}\binom{\theta^{x}}{\eta^{x}}_{q^{\frac{1}{2}}}q^{-\left(N_{x-1}^{-}(\boldsymbol{\theta})+\frac{1}{2}\theta^{x}\right)(\xi^{x}+\eta^{x})},\end{array}$
(34)

where in the second equation we used the following identities:

$\left(pq^{N(\boldsymbol{\eta})+1}\right)_{\infty}=(pq)_{\infty}(pq^{N(\boldsymbol{\eta})+1})_{-N(\boldsymbol{\eta})}=(pq)_{\infty}\frac{(-1)^{N(\boldsymbol{\eta})}q^{-\binom{N(\boldsymbol{\eta})+1}{2}}}{p^{N(\boldsymbol{\eta})}(p^{-1}q^{-N(\boldsymbol{\eta})})_{N(\boldsymbol{\eta})}},$
(35)
$\frac{(p^{-1}q^{-N(\boldsymbol{\theta}-\boldsymbol{\xi})})_{N(\boldsymbol{\theta}-\boldsymbol{\xi})}}{(p^{-1}q^{-N(\boldsymbol{\eta})})_{N(\boldsymbol{\eta})}}=\frac{(p^{-1}q^{-N(\boldsymbol{\theta}-\boldsymbol{\xi})})_{\infty}}{(p^{-1}q^{-N(\boldsymbol{\eta})})_{\infty}}.$
(36)

Note that if we substitute $q$ by $q^{2}$ and let $p=\alpha^{-1}q^{-1}$ we get
exactly (15), i.e. the orthogonal duality function for single species case.

∎

Now we work on the orthogonality of
$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})$.
Fix two particle configurations $\boldsymbol{\eta}$ and
$\bar{\boldsymbol{\eta}}$. Let $\boldsymbol{\xi}$ be another particle
configuration which is viewed as a variable, rather than a fixed
configuration. As in section 3.1, define the intermediate particle
configurations

$\boldsymbol{\zeta}^{(1)},\ldots,\boldsymbol{\zeta}^{(n-1)},\quad\quad\bar{\boldsymbol{\zeta}}^{(1)},\ldots,\bar{\boldsymbol{\zeta}}^{(n-1)}.$

Since $\boldsymbol{\xi}_{i}$ is independent of $\boldsymbol{\zeta}^{(i)}_{i}$,
thus sum from left to right $\boldsymbol{\xi}_{0}$ to
$\boldsymbol{\xi}_{n-1}$, and apply Lemma 4.1,

$\prod_{i=0}^{n-1}\sum_{\boldsymbol{\xi}_{i}}\frac{w_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})}{h^{1/2}_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})h^{1/2}_{\boldsymbol{\theta}^{(i)}}(\bar{\boldsymbol{\zeta}}^{(i)}_{i})}K^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i})K^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i},\bar{\boldsymbol{\zeta}}^{(i)}_{i})=\delta_{\boldsymbol{\eta},\bar{\boldsymbol{\eta}}},$
(37)

Notice that $\boldsymbol{\theta}^{(i)}=\bar{\boldsymbol{\theta}}^{(i)}$ in the
summation of $\boldsymbol{\xi}_{i}$, thus we have

$\sum_{\boldsymbol{\xi}}\prod_{i=0}^{n-1}\sqrt{\frac{w_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}}K^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i})\sqrt{\frac{w_{\bar{\boldsymbol{\theta}}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\bar{\boldsymbol{\theta}}^{(i)}}(\bar{\boldsymbol{\zeta}}^{(i)}_{i})}}K^{\bar{\boldsymbol{\theta}}^{(i)}}(\boldsymbol{\xi}_{i},\bar{\boldsymbol{\zeta}}^{(i)}_{i})=\delta_{\boldsymbol{\eta},\bar{\boldsymbol{\eta}}},$
(38)

or, in other words,
$\prod_{i=0}^{n-1}\sqrt{\frac{w_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}}K^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i})$
is orthonormal w.r.t standard inner product.

Thus far, the parameter $\boldsymbol{\alpha}$ has not come up. Now, if we
substitute $q$ by $q^{2}$, let $p_{i}=\alpha_{i}^{-1}q^{-1}$ and divide by
$\sqrt{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\boldsymbol{\eta})}$ , up to a
constant, we get
$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})$,
which is the orthogonal duality function for multi-species case. More
precisely, now $w$ and $h$ depend on $\boldsymbol{\alpha}$ and we write

$\sum_{\boldsymbol{\xi}}\prod_{i=0}^{n-1}\sqrt{\frac{w_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}}\frac{K^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i})}{\sqrt{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\boldsymbol{\eta})}}\sqrt{\frac{w_{\bar{\boldsymbol{\theta}}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\bar{\boldsymbol{\theta}}^{(i)}}(\bar{\boldsymbol{\zeta}}^{(i)}_{i})}}\frac{K^{\bar{\boldsymbol{\theta}}^{(i)}}(\boldsymbol{\xi}_{i},\bar{\boldsymbol{\zeta}}^{(i)}_{i})}{\sqrt{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\bar{\boldsymbol{\eta}})}}\mu^{n}(\boldsymbol{\xi})=\frac{\delta_{\boldsymbol{\eta},\bar{\boldsymbol{\eta}}}}{\mu^{n}(\boldsymbol{\eta})}.$
(39)

By a direct, but lengthy calculation,

$\prod_{i=0}^{n-1}\sqrt{\frac{w_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})}{h_{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}}=\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})\sqrt{\prod_{i=0}^{n-1}\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}$

and similarly for
$\mathcal{C}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\bar{\boldsymbol{\eta}}).$
Since

$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})=\mathcal{G}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})\prod_{i=0}^{n-1}D^{\boldsymbol{\theta}^{(i)}}_{\alpha_{i}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i}),$

where

$\mathcal{G}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})=\sqrt{\frac{\prod_{i=0}^{n-1}\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\zeta}^{(i)}_{i})}{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\boldsymbol{\eta})}},$

this completes the proof.

###  4.2 Previously Known Results

####  4.2.1 A general algebraic set up for duality

The multi–species ASEP$(q,\boldsymbol{\theta})$ was constructed in [Kua17],
generalizing the single–species ASEP$(q,\theta)$ in [CGRS16]. In this process,
up to $\theta$ particles can occupy each site, and the particles have a drift
with asymmetry parameter $q^{\theta}$. The generator was constructed using the
Casimir element from [KLLPZ21]. An inhomogeneous version was constructed in
[Kua19], where the maximum number of particles per site can vary.

The set-up involves the following ingredients see Section 3.5 of [Kua17] or
Proposition 2.1 of [CGRS16] for more details:

  * •

A Hamiltonian $H$, and a symmetry $S$ as in Definition 2.2

  * •

A change-of-basis diagonal matrix $B$, such that $B^{-1}HB$ is self–adjoint.

  * •

A ground state transformation $G$, which is a diagonal matrix such that
$L:=G^{-1}HG$ is the generator of a Markov process and $\tilde{S}=G^{-1}SG$ is
a symmetry for $L$.

The result is then that $D:=\tilde{S}G^{2}B^{-2}=G^{-1}SG^{-1}B^{2}$ is a
self–duality function, and $G^{2}B^{-2}$ are reversible measures for the
multi–species ASEP$(q,\boldsymbol{\theta})$. More specifically,

$G^{2}B^{-2}(\boldsymbol{\xi},\boldsymbol{\xi})=\sum_{\boldsymbol{k}}\frac{\mu_{\boldsymbol{k}}^{n}(\boldsymbol{\xi})}{Z_{n,\boldsymbol{k}}},$
(40)

for some normalizing constants $Z_{n,\boldsymbol{k}}$. The matrices $G$ and
$B$ are not unique, but any differences can be absorbed into the normalizing
constants.

###  4.3 Preliminary Lemmas

In order for the symmetries to be called unitary, they need to be unitary with
respect to an inner product on the representation. The next lemma explicitly
defines such an inner product.

######  Lemma 4.2.

(a) Define an inner product on $V_{m}^{(n)}$ by

$\langle
v_{\mu},v_{\nu}\rangle=1_{\\{\mu=\nu\\}}q^{O(\mu)}\frac{[\mu_{0}]_{q}^{!}\cdots[\mu_{n}]_{q}^{!}}{[m]_{q}^{!}},$

where

$O(\mu)=-(\mu_{1}+2\mu_{2}+\ldots+n\mu_{n})+\frac{\mu_{0}^{2}}{2}+\frac{\mu_{1}^{2}}{2}+\cdots+\frac{\mu_{n}^{2}}{2}.$

Then $V_{m}^{(n)}$ is a $*$–representation of
$\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$ with action given by (13).

(b) An equivalent formula for the inner product is

$\langle v_{\mu},v_{\nu}\rangle=\mathrm{const}\cdot
1_{\\{\mu=\nu\\}}q^{(\rho,\mu)}\\{\mu_{0}\\}_{q^{2}}^{!}\cdots\\{\mu_{n}\\}_{q^{2}}^{!},$

where $\rho$ is half the sum of the positive roots and $(\cdot,\cdot)$ is the
Killing form.

(c) For any $X\in\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$, let $M_{X}$ be the
matrix of the action of $X$ on $V_{m}^{(n)}$ with respect to the basis
$\\{v_{\mu}:\mu\in\mathcal{B}_{m}^{(n)}\\}$. Then

$B^{2}M_{X}^{T}B^{-2}=M_{X^{*}},$

where $B$ is the diagonal matrix with entries

$B(\lambda,\mu)=1_{\\{\lambda=\mu\\}}\frac{1}{\sqrt{\\{\mu_{0}\\}_{q^{2}}^{!}\cdots\\{\mu_{n}\\}_{q^{2}}^{!}}}.$

In particular, if $U$ is unitary i.e. $U^{*}=U^{-1}$, then

$B^{2}M_{U}^{T}B^{-2}=M_{U}^{-1}.$

###### Proof.

(a) This part of the result has possibly appeared before, but the authors were
unable to find it in the literature. In any case, it is not difficult to prove
anyway, so for completeness the proof is presented here.

Recall that the $*$ involution is given on generators by

$E_{i,i+1}^{*}=E_{i+1,i}q^{E_{ii}-E_{i+1,i+1}},\quad
E_{i+1,i}^{*}=q^{-E_{ii}+E_{i+1,i+1}}E_{i,i+1},\quad(q^{E_{ii}})^{*}=q^{E_{ii}}.$

So one checks that

$\displaystyle\langle E_{i,i+1}v_{\mu},v_{\nu}\rangle$
$\displaystyle=[\mu_{i+1}]_{q}\langle v_{\mu+\epsilon_{i}},v_{\nu}\rangle$
$\displaystyle=1_{\\{\mu+\epsilon_{i}=\nu\\}}q^{O(\nu)}[\mu_{i+1}]_{q}\frac{[\nu_{0}]_{q}^{!}\cdots[\nu_{n}]_{q}^{!}}{[m]_{q}^{!}}$
$\displaystyle=1_{\\{\mu+\epsilon_{i}=\nu\\}}[\mu_{i}+1]_{q}\cdot
q^{O(\nu)}\frac{[\mu_{0}]_{q}^{!}\cdots[\mu_{n}]_{q}^{!}}{[m]_{q}^{!}}$

while

$\displaystyle\langle v_{\mu},E_{i+1,i}q^{E_{ii}-E_{i+1,i+1}}v_{\nu}\rangle$
$\displaystyle=[\nu_{i}]_{q}q^{\nu_{i}-\nu_{i+1}}\langle
v_{\mu},v_{\nu-\epsilon_{i}}\rangle$
$\displaystyle=1_{\\{\mu=\nu-\epsilon_{i}\\}}[\nu_{i}]_{q}q^{\nu_{i}-\nu_{i+1}}q^{O(\mu)}\frac{[\mu_{0}]_{q}^{!}\cdots[\mu_{n}]_{q}^{!}}{[m]_{q}^{!}}.$

Now observe that when $\nu=\mu+\epsilon_{i}$, we have $\nu_{i}=\mu_{i}+1$ and

$\displaystyle\nu_{i}-\nu_{i+1}+O(\mu)$ $\displaystyle=\left(\sum_{k\neq
i,i+1}\left(-k\mu_{k}+\frac{\mu_{k}^{2}}{2}\right)\right)+\mu_{i}+1-\mu_{i+1}+1-i\mu_{i}-(i+1)\mu_{i+1}+\frac{\mu_{i}^{2}}{2}+\frac{\mu_{i+1}^{2}}{2}$
$\displaystyle=\left(\sum_{k\neq
i,i+1}\left(-k\nu_{k}+\frac{\nu_{k}^{2}}{2}\right)\right)+1-i(\nu_{i}-1)-(i+1)(\nu_{i+1}+1)+\frac{(\mu_{i}+1)^{2}}{2}+\frac{(\mu_{i+1}-1)^{2}}{2}$
$\displaystyle=O(\nu).$

Thus $\langle E_{i,i+1}v_{\mu},v_{\nu}\rangle=\langle
v_{\mu},E_{i,i+1}^{*}v_{\nu}\rangle$ for all
$\mu,\nu\in\mathcal{B}_{m}^{(n)}$. One can similarly verify that …. …

(b) The $[m]_{q}^{!}$ in the denominator can be removed, but by leaving it in
the inner product can be written as the multinomial

$\langle
v_{\mu},v_{\nu}\rangle=1_{\\{\mu=\nu\\}}\frac{q^{O(\mu)}}{\binom{m}{\mu_{0},\ldots,\mu_{n}}_{q}}.$

Letting $\rho$ be half the sum of the positive roots, we have

$(\rho-\frac{n}{2}I,\mu)=-\mu_{1}-2\mu_{2}-\ldots-(n-1)\mu_{n-1}-n\mu_{n}$

where $(\cdot,\cdot)$ is the Killing form. This shows the equivalence of the
two formulas.

(c) By definition,

$\langle Xv_{\mu},v_{\lambda}\rangle=\langle v_{\mu},X^{*}v_{\lambda}\rangle$

so therefore

$M_{X}(\lambda,\mu)\langle
v_{\lambda},v_{\lambda}\rangle=M_{X^{*}}(\mu,\lambda)\langle
v_{\lambda},v_{\lambda}\rangle,$

implying the result. The statement when $U$ is unitary is immediate.

∎

This paper uses an inhomogeneous multi–spceies ASEP$(q,\boldsymbol{\theta})$,
where the maximum number of particles at a site may vary. The next proposition
establishes that the algebraic construction of [Kua17] still carries over in
the inhomogeneous case. Compared to [Kua17], the argument below is slightly
stremalined by using the $*$–algebra sturcture.

######  Proposition 4.3.

Consider the representation

$V_{\theta_{1}}^{(n)}\otimes\cdots\otimes V_{\theta_{L}}^{(n)}$

of the quantum group $\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$.

The generator of the multi–species ASEP$(q,\vec{m})$ can be written as

$G^{-1}\Delta^{L-1}(C_{1})G,$

where $G$ is the diagonal map with entries

$\prod_{i=0}^{n}\prod_{x=1}^{L}\frac{1}{\left[\eta_{i}^{x}\right]_{q}^{!}}\times\prod_{1\leq
y<x\leq L}\prod_{i=0}^{n-1}q^{-\eta_{i+1}^{y}\eta_{[1,i]}^{x}}.$ (41)

###### Proof.

The proofs of Propositions 4.1 and 4.2 of [Kua17] generalize to the
inhomogeneous case, which would imply the proposition here. By using the
$*$–algebra structure, the proof presented below is slightly more streamlined
than the proofs in [Kua17]. The reader does not need to refer to [Kua17] for
the proof below.

Fix $0\leq i<j\leq n$. The claim is that

$h^{x,x+1}\left(\left(\mu_{i\rightarrow j},\lambda_{j\rightarrow
i}\right),(\mu,\lambda)\right)\\\
=q^{-2}q^{\mu_{i}+\mu_{i+1}+\cdots+\mu_{j-1}}\left\\{\mu_{i}\right\\}_{q^{2}}q^{\lambda_{i+1}+\cdots+\lambda_{j}}\left\\{\lambda_{j}\right\\}_{q^{2}}\cdot
q^{2\left(\lambda_{j+1}+\ldots+\lambda_{n}\right)}\cdot
q^{2\left(\mu_{1}+\ldots+\mu_{i-1}\right)}$ (42)

and

$h^{x,x+1}\left(\left(\mu_{j\rightarrow i},\lambda_{i\rightarrow
j}\right),(\mu,\lambda)\right)=q^{\mu_{i}+\ldots+\mu_{j-1}}\left\\{\mu_{j}\right\\}_{q^{2}}q^{\lambda_{i+1}+\ldots+\lambda_{j}}\left\\{\lambda_{i}\right\\}_{q^{2}}\cdot
q^{2\left(\lambda_{j+1}+\ldots+\lambda_{n}\right)}\cdot
q^{2\left(\mu_{1}+\ldots+\mu_{i-1}\right)}$ (43)

First start with the equation (42), which is the equation for left jumps. When
applying the co–product to the central element $C_{1}$, the number of
off–diagonal terms equals

$\sum_{1\leq i<j\leq n}\left((j-i+1)^{2}-(j-i+1)\right)=\sum_{1\leq i<j\leq
n}(2(i-1)(n-j)+2+2(n-j)+2(i-1))$ (44)

Each of the four summands on the right–hand–side corresponds to four
expressions of the form

$E_{ri}E_{jr}\otimes E_{is}E_{sj},\quad E_{ji}\otimes E_{ij},\quad
E_{ri}E_{jr}\otimes E_{ij},\quad E_{ij}\otimes E_{is}E_{sj}.$ (45)

Therefore the term that needs to be calculated can be expressed as a sum of
four types of terms:

$q^{2j+2}q^{2E_{ii}}E_{ji}\otimes
q^{2E_{jj}}E_{ij}+\left(q-q^{-1}\right)\sum_{s=j+1}^{n}q^{2s+1}q^{2E_{ii}}E_{ji}\otimes
q^{E_{ss}}E_{is}q^{E_{jj}}E_{sj}\\\
+\left(q-q^{-1}\right)q^{2j+2}\sum_{r=1}^{i-1}q^{E_{ii}}E_{ri}q^{E_{rr}}E_{jr}\otimes
q^{2E_{jj}}E_{ij}+\left(q-q^{-1}\right)^{2}\sum_{r=1}^{i-1}\sum_{s=j+1}^{n}q^{2s+1}q^{E_{ii}}E_{ri}q^{E_{rr}}E_{jr}\otimes
q^{E_{ss}}E_{is}q^{E_{jj}}E_{sj}.$ (46)

At this point, the right jumps can be found with a similar calculation, which
is how [Kua17] proceeded with the proof. Actually, with the $*$–bialgebra
structure, the right jumps can be found much more quickly. Since
$\Delta(C_{1})=\Delta(C_{1}^{*})=\Delta(C_{1})^{*}$ this implies that

$\langle\Delta(C_{1})(v_{\mu_{i\rightarrow j}}\otimes v_{\lambda_{j\rightarrow
i}}),v_{\mu}\otimes v_{\lambda}\rangle=\langle v_{\mu_{i\rightarrow j}}\otimes
v_{\lambda_{j\rightarrow i}},\Delta(C_{1})(v_{\mu}\otimes
v_{\lambda})\rangle,$

which means that

$h^{x,x+1}\left((\mu,\lambda),\left(\mu_{i\rightarrow j},\lambda_{j\rightarrow
i}\right)\right){\|v_{\mu}\otimes
v_{\lambda}\|^{2}}=h^{x,x+1}\left(\left(\mu_{i\rightarrow
j},\lambda_{j\rightarrow i}\right),(\mu,\lambda)\right){\|v_{\mu_{i\rightarrow
j}}\otimes v_{\lambda_{j\rightarrow i}}\|^{2}}$

so thus

$h^{x,x+1}\left((\mu,\lambda),\left(\mu_{j\rightarrow i},\lambda_{i\rightarrow
j}\right)\right)\\\
=q^{-2}q^{\mu_{i}+\mu_{i+1}+\cdots+\mu_{j-1}}\left\\{\mu_{i}\right\\}_{q^{2}}q^{\lambda_{i+1}+\cdots+\lambda_{j}}\left\\{\lambda_{j}\right\\}_{q^{2}}\cdot
q^{2\left(\lambda_{j+1}+\ldots+\lambda_{n}\right)}\cdot
q^{2\left(\mu_{1}+\ldots+\mu_{i-1}\right)}\times\frac{\\{\mu_{j}+1\\}_{q^{2}}}{\\{\mu_{i}\\}_{q^{2}}}\frac{\\{\lambda_{i}+1\\}_{q^{2}}}{\\{\lambda_{j}\\}_{q^{2}}}.$

Setting $\hat{\mu}=\mu_{j\rightarrow i},\hat{\lambda}=\lambda_{i\rightarrow
j}$, so that $\mu_{i}=\hat{\mu}_{i}+1,\lambda_{j}=\hat{\lambda}_{j}+1$ shows
that (46) holds.

The remaining steps with the ground state transformation $G$ are identical to
[Kua17], and will not be repeated here.

∎

###  4.4 Identifying the unitary symmetry from the $*$–bialgebra structure

Given the one-to-one link between self-duality functions and the corresponding
symmetries as shown in Section 2.4, it is natural to ask which is the symmetry
associated to the orthogonal self-duality function of the multi–species
ASEP$(q,\boldsymbol{\theta})$. Since the aimed symmetry, introduced in the
next lemma, is associated to an orthogonal polynomial then we know it must be
a unitary operator as the norm of the cheap self-duality function is
preserved. For any $X,Y\in\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$,

$\left(e_{q^{2}}(X)\mathcal{E}_{q^{2}}(Y)\right)^{-1}=e_{q^{2}}(-Y)\mathcal{E}_{q^{2}}(-X),$

because the $e$ and the $\mathcal{E}$ are inverses (9), while

$\left(e_{q^{2}}(X)\mathcal{E}_{q^{2}}(Y)\right)^{*}=\mathcal{E}_{q^{2}}(Y^{*})e_{q^{2}}(X^{*}),$

therefore

$\left(e_{q^{2}}(X)\mathcal{E}_{q^{2}}(Y)\right)^{-1}=\left(e_{q^{2}}(X)\mathcal{E}_{q^{2}}(Y)\right)^{*}\Longleftrightarrow
e_{q^{2}}(-Y^{*})e_{q^{2}}(-Y)=e_{q^{2}}(X^{*})e_{q^{2}}(X).$

######  Lemma 4.4.

Define the element $U(\lambda)$ by

$U(\lambda):={e}_{q^{2}}^{1/2}\left(-\gamma\lambda
q^{2E_{ii}}\right){e}_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)\\\
\times\mathcal{E}_{q^{2}}(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\mathcal{E}_{q^{2}}^{1/2}\left(\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}}\right),$

where $\lambda=\gamma(1-q^{2})(q-q^{-1})$. Then $U(\lambda)$ is unitary, in
other words $U(\lambda)^{*}=U(\lambda)^{-1}$.

###### Proof.

This is modified from the proof in [GPVYZ16] to avoid $q$–oscillator algebras
and be more applicable to other semi-simple Lie algebras.

Substituting the value of $\lambda$, we have that

$U^{*}=\mathcal{E}_{q^{2}}^{1/2}\left(\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}}\right)\mathcal{E}_{q^{2}}\left(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1}\right)^{*}\times e_{q^{2}}(\lambda
E_{i+1,i}q^{E_{ii}})^{*}e_{q^{2}}^{1/2}\left(-\gamma\lambda
q^{2E_{ii}}\right)$

and

$U^{-1}=e_{q^{2}}^{1/2}\left(-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}}\right){e}_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\\\
\times\mathcal{E}_{q^{2}}\left(-\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)\mathcal{E}_{q^{2}}^{1/2}\left(\gamma\lambda
q^{2E_{ii}}\right).$

Therefore the equality $U^{*}=U^{-1}$ is equivalent to

$\mathcal{E}_{q^{2}}\left(-\gamma(1-q^{2})(q-q^{-1})q^{E_{i+1,i+1}}E_{i,i+1}\right)^{*}e_{q^{2}}(\lambda
E_{i+1,i}q^{E_{ii}})^{*}e_{q^{2}}\left(-\gamma\lambda q^{2E_{ii}}\right)\\\
=e_{q^{2}}\left(-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}}\right){e}_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\mathcal{E}_{q^{2}}\left(-\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right),$

By the $*$–structure on $\mathcal{U}_{q}(\mathfrak{sl}_{n+1})$, this is
equivalent to

$\mathcal{E}_{q^{2}}\left(-\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})e_{q^{2}}\left(-\gamma\lambda q^{2E_{ii}}\right)\\\
=e_{q^{2}}\left(-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}})\right){e}_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\mathcal{E}_{q^{2}}\left(-\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right),$

Rewriting the identity solely in terms of the $q$–exponentials $e_{q^{2}}$, it
suffices to prove

$e_{q^{2}}(\lambda q^{E_{i+1,i+1}}E_{i,i+1})e_{q^{2}}\left(-\gamma\lambda
q^{2E_{ii}}\right)e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)\\\
=e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)e_{q^{2}}\left(-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}})\right){e}_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1}),$

As pointed out in section 2.3 of [GPVYZ16], the $q$–BCH equations imply

$\displaystyle e_{q}\left(\alpha
A_{-}B_{+}\right)e_{q}\left(\frac{\alpha\beta}{(1-q)^{2}}q^{B_{0}}\right)e_{q}\left(\beta
A_{+}B_{-}\right)$ $\displaystyle=e_{q}\left(\beta
A_{+}B_{-}\right)e_{q}\left(\frac{\alpha\beta}{(1-q)^{2}}q^{A_{0}}\right)e_{q}\left(\alpha
A_{-}B_{+}\right)$ (47) $\displaystyle\mathcal{E}_{q}\left(\gamma
A_{+}B_{-}\right)\mathcal{E}_{q}\left(-\frac{\gamma\delta}{(1-q)^{2}}q^{B_{0}}\right)\mathcal{E}_{q}\left(\delta
A_{-}B_{+}\right)$ $\displaystyle=\mathcal{E}_{q}\left(\delta
A_{-}B_{+}\right)\mathcal{E}_{q}\left(-\frac{\gamma\delta}{(1-q)^{2}}q^{A_{0}}\right)\mathcal{E}_{q}\left(\gamma
A_{+}B_{-}\right)\ $ (48)

for any $\alpha,\beta,\gamma,\delta$. Their proof uses the Schwinger model for
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$, where the generators are written in
terms of two copies of the $q$–oscillator algebra. In order to avoid
needlessly introducing additional algebraic structure, and because this paper
uses a different $*$–structure, the calculation will be repeated here.

We have that

$\displaystyle e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1},)E_{i+1,i}q^{E_{ii}}\mathcal{E}_{q^{2}}(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1},)$
$\displaystyle=E_{i+1,i}q^{E_{ii}}-\lambda\frac{q^{2E_{i+1,i+1}}}{q-q^{-1}}$
(49)
$\displaystyle+\sum_{n=0}^{\infty}\lambda^{n+1}q^{n(n-5)/2}\frac{{q^{nE_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n}}}{(1-q^{2})(q-q^{-1})},$
(50) $\displaystyle e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1},)q^{2E_{ii}}\mathcal{E}_{q^{2}}(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1},)$
$\displaystyle=\sum_{n=0}^{\infty}\lambda^{n}q^{n(n-5)/2}{q^{nE_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n}}.$
(51)

To see equation (49), first calculate, using (7) with $X=E_{i+1,i}$ and
$Y=E_{i,i+1}$,

$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},E_{i+1,i}q^{E_{ii}}]^{\prime}_{0}$
$\displaystyle=E_{i+1,i}q^{E_{ii}},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},E_{i+1,i}q^{E_{ii}}]^{\prime}_{1}$
$\displaystyle=\frac{q^{2E_{ii}}-q^{2E_{i+1,i+1}}}{q-q^{-1}},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},E_{i+1,i}q^{E_{ii}}]^{\prime}_{n}$
$\displaystyle=\frac{q^{(n-1)E_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n-1}(q^{-2}-q^{2})(q^{-1}-q^{5})\cdots(q^{n-4}-q^{3n-4})}{q-q^{-1}},$
$\displaystyle=\frac{q^{-2-1+0+1+\ldots+(n-4)}q^{(n-1)E_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n-1}(q^{2};q^{2})_{n}}{(1-q^{2})(q-q^{-1})},\quad
n\geq 2.$

Since $-2+(-1)+0+1+\ldots+(n-3)=(n-2)(n-3)/2-3=n^{2}/2-5n/2$, therefore (49)
holds. To see (51), again calculate

$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{2E_{ii}}]^{\prime}_{0}$
$\displaystyle=q^{2E_{ii}},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{2E_{ii}}]_{1}^{\prime}$
$\displaystyle=(q^{-2}-1)q^{E_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{2E_{ii}}]_{n}^{\prime}$
$\displaystyle=(q^{-2}-1)(q^{-1}-q^{3})(1-q^{6})\cdots(q^{n-3}-q^{3n-3})q^{nE_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n}$
$\displaystyle=q^{-2}q^{-1}q^{0}\cdots
q^{n-3}(q^{2};q^{2})_{n}q^{nE_{i+1,i+1}}q^{2E_{ii}}E_{i,i+1}^{n},\quad\quad
n\geq 2.$

Note that (51) can also be calculated by first noting that

$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{-2E_{ii}}]^{\prime}_{0}$
$\displaystyle=q^{-2E_{ii}},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{-2E_{ii}}]_{1}^{\prime}$
$\displaystyle=(q^{2}-1)q^{E_{i+1,i+1}}q^{-2E_{ii}}E_{i,i+1},$
$\displaystyle[q^{E_{i+1,i+1}}E_{i,i+1},q^{2E_{ii}}]_{n}^{\prime}$
$\displaystyle=0,\quad\quad n\geq 2.$

and then recalling that the symmetry is in fact an automorphism, results in
the formal series

$\displaystyle\left(e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})q^{-2E_{ii}}\mathcal{E}_{q^{2}}(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\right)^{-1}$
$\displaystyle=\left(q^{-2E_{ii}}(1-\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\right)^{-1}$
$\displaystyle=\sum_{n=0}^{\infty}\left(\lambda
q^{E_{i+1,i+1}}E_{i,i+1}\right)^{n}q^{2E_{ii}}$
$\displaystyle=\sum_{n=0}^{\infty}\lambda^{n}q^{2E_{ii}}q^{-2n}q^{n(n-1)/2}q^{nE_{i+1,i+1}}E_{i,i+1}^{n}.$

So combining (49) and (51), and applying them to the element
$(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}-\lambda q^{2E_{ii}}$, all
$\lambda^{n}$ terms cancel for $n\geq 2$, leaving only:

$e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\left((1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}-\lambda
q^{2E_{ii}}\right)\mathcal{E}_{q^{2}}(\lambda q^{E_{i+1,i+1}}E_{i,i+1})\\\
=\left(E_{i+1,i}q^{E_{ii}}-\lambda\frac{q^{2E_{i+1,i+1}}}{q-q^{-1}}\right)(1-q^{2})(q-q^{-1}),$

implying that

$e_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}-\gamma\lambda
q^{2E_{ii}}\right)\mathcal{E}_{q^{2}}(\lambda q^{E_{i+1,i+1}}E_{i,i+1})\\\
=e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}})\right).$

By (8),

$e_{q^{2}}(\lambda q^{E_{i+1,i+1}}E_{i,i+1})e_{q^{2}}\left(-\gamma\lambda
q^{2E_{ii}}\right)e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)\mathcal{E}_{q^{2}}(\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\\\
=e_{q^{2}}\left(\gamma(1-q^{2})(q-q^{-1})E_{i+1,i}q^{E_{ii}}\right)e_{q^{2}}\left(-\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}})\right),$

which implies the desired identity.

∎

As proven in [Kua17], the diagonal matrix $G^{2}B^{-2}$ encodes reversible
measures for the multi–species inhomogeneous ASEP$(q,\boldsymbol{\theta})$.
Note that the $B$ in [Kua17] is the same matrix $B$ in the current paper.
There is likely an algebraic reason for this. By defining a diagonal matrix
$A$ which is constant on irreducible components of the interacting particle
system, one can obtain a family of reversible measures given by
$AG^{2}B^{-2}$.

The next theorem generalizes (in a sense) Theorem 3.2 and equation (95) of
[CFG21].

######  Theorem 4.5.

(i) The function $D$ given by

$D:=G^{-1}M_{U(\lambda)}G^{-1}B^{2}$

is a self–duality function, and the entries are orthogonal with respect to the
reversible measures, in the sense that

$D^{T}G^{2}B^{-2}D=G^{-2}B^{2}.$

(ii) More generally, let $A$ be a quantity which is conserved under the
dynamics; in other words, $A$ is a diagonal matrix such that $LA=AL$. Then the
function $D_{A}$ given by

$D_{A}:=AG^{-1}M_{U(\lambda)}G^{-1}B^{2}A$

is a self–duality function, and its entries are orthogonal with respect to the
reversible measures, in the sense that

$D_{A}^{T}A^{-2}G^{2}B^{-2}D_{A}=A^{2}G^{-2}B^{2}.$

###### Proof.

(i) The fact that $D$ is a self–duality function follows from the general
framework of [Kua17].

To see the second statement, note first that

$\displaystyle D^{T}G^{2}B^{-2}D$
$\displaystyle=B^{2}G^{-1}M_{U(\lambda)}^{T}G^{-1}G^{2}B^{-2}G^{-1}M_{U(\lambda)}G^{-1}B^{2}$
$\displaystyle=B^{2}G^{-1}M_{U(\lambda)}^{T}B^{-2}M_{U(\lambda)}G^{-1}B^{2}.$

Since $U(\lambda)$ is unitary, then by part (c) of Lemma 4.2,

$D^{T}G^{2}B^{-2}D=B^{2}G^{-1}B^{-2}M_{U(\lambda)}^{-1}M_{U(\lambda)}G^{-1}B^{2},$

which simplifies to $G^{-2}B^{2}$, as needed.

(ii) Noting that $D_{A}=ADA$ where $D$ is from part (i), we thus have

$LD_{A}=LADA=ALDA=ADL^{T}A=ADAL^{T}=D_{A}L^{T},$

so $D_{A}$ is also a self–duality function.

Since $D^{T}G^{2}B^{-2}D=G^{-2}B^{2}$, we have

$D_{A}^{T}A^{-2}G^{2}B^{-2}D_{A}=AD^{T}A^{T}A^{-2}G^{2}B^{-2}ADA=AG^{-2}B^{2}A=A^{2}G^{-2}B^{2}.$

∎

###  4.5 Proof of Theorem 3.1: Calculation of Orthogonal Duality Functions

With the algebraic machinery, the explicit form of the orthogonal duality
functions can now be calculated. The duality function is given by Theorem 4.5,
and is written as:

$D_{A}:=AG^{-1}M_{U(\lambda)}G^{-1}B^{2}A$

and it satisfies

$D_{A}^{T}A^{-2}G^{2}B^{-2}D_{A}=A^{2}G^{-2}B^{2}.$

Recall that

$U_{i}(\lambda):={e}_{q^{2}}^{1/2}\left(-\gamma\lambda
q^{2E_{ii}}\right){e}_{q^{2}}\left(\lambda
E_{i+1,i}q^{E_{ii}}\right)\mathcal{E}_{q^{2}}(-\lambda
q^{E_{i+1,i+1}}E_{i,i+1})\mathcal{E}_{q^{2}}^{1/2}\left(\gamma\lambda(1-q^{2}){q^{2E_{i+1,i+1}}}\right),$

and $G$ is the diagonal matrix with entries

$\prod_{i=0}^{n}\prod_{x=1}^{L}\frac{1}{\left[\eta_{i}^{x}\right]_{q}^{!}}\times\prod_{1\leq
y<x\leq L}\prod_{i=0}^{n-1}q^{-\eta_{i+1}^{y}\eta_{[1,i]}^{x}}$

and $B$ is the diagonal matrix with entries with entries

$B(\lambda,\mu)=1_{\\{\lambda=\mu\\}}\frac{1}{\sqrt{\\{\mu_{0}\\}_{q^{2}}^{!}\cdots\\{\mu_{n}\\}_{q^{2}}^{!}}}.$

####  4.5.1 Inhomogeneous, single–species case

First, briefly note that the $e_{q^{2}}^{1/2}$ and $\mathcal{E}_{q^{2}}^{1/2}$
terms are actually constants in the duality function and can be ignored.

The next step is to show that in the case of a fixed $i$, the duality function
is an inhomogeneous version of the one from [CFG21]. Doing so is essentially
matching notation from this paper to the notation in [CFG21], rather than
developing new proofs. That previous paper [CFG21] uses generators of
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$:

$\begin{array}[]{l}q^{A^{0}}A^{+}=qA^{+}q^{A^{0}}\\\
q^{A^{0}}A^{-}=q^{-1}A^{-}q^{A^{0}}\\\
{\left[A^{+},A^{-}\right]=\left[2A^{0}\right]_{q}}\end{array}$

Additionally, the representation

$\displaystyle A^{+}|n\rangle$
$\displaystyle=\sqrt{[\theta-n]_{q}[n+1]_{q}}|n+1\rangle$ $\displaystyle
A^{-}|n\rangle$ $\displaystyle=\sqrt{[n]_{q}[\theta-n+1]_{q}}|n-1\rangle$
$\displaystyle A^{0}|n\rangle$ $\displaystyle=(n-\theta/2)|n\rangle$

needs to be compared with

$\displaystyle E_{i,i+1}v_{\mu}$
$\displaystyle=[\mu_{i+1}]_{q}v_{\mu+\epsilon_{i}},\quad 0\leq i<n$
$\displaystyle E_{i+1,i}v_{\mu}$
$\displaystyle=[\mu_{i}]_{q}v_{\mu-\epsilon_{i}},\quad 0\leq i<n$
$\displaystyle q^{E_{ii}}v_{\mu}$ $\displaystyle=q^{\mu_{i}}v_{\mu},\quad
0\leq i\leq n$

Now, note that

$A^{-}A^{+}|n\rangle=\sqrt{[\theta-n]_{q}[n+1]_{q}}A^{-}|n+1\rangle={[\theta-n]_{q}[n+1]_{q}}|n\rangle,$

while in the one–species case

$E_{01}E_{10}v_{\mu}=[\mu_{0}]_{q}[\mu_{1}+1]_{q}v_{\mu}.$

Thus it is natural to replace $A^{+}$ with $E_{10}$ and $A^{-}$ with $E_{01}$.

In the notation of section 8.3 from [CFG21], there are the operators

$S_{\alpha}=e_{q^{2}}\left(\sqrt{\alpha}\left(1-q^{2}\right)\Delta^{L-1}\left(q^{A^{0}}A^{+}\right)\right)$
(52)

and

$\hat{S}_{\alpha}=\mathcal{E}_{q^{2}}\left(\sqrt{\alpha}\left(1-q^{2}\right)q^{-\theta/2-\theta
L}\Delta^{L-1}\left(q^{-A^{0}}A^{+}\right)\right).$ (53)

However, the present paper does not use $q^{A_{0}}$ because it is not an
element of the quantum group. In the notation of this paper, $q^{A_{0}}$ is
the analog of $q^{(E_{00}-E_{11})/2}$. Because the element $q^{E_{00}+E_{11}}$
is central, inserting it into the $q$–exponentials will introduce an
irrelevant constant, in the sense that the constant is preserved under the
dynamics. The constant $q^{-\theta/2-\theta L}$ is similarly meaningless.
Therefore, the operators can be replaced with

$\mathfrak{S}_{\alpha}=e_{q^{2}}\left(\sqrt{\alpha}\left(1-q^{2}\right)\Delta^{L-1}\left(E_{10}q^{E_{00}}\right)\right)$
(54)

and

$\hat{\mathfrak{S}}_{\alpha}=\mathcal{E}_{q^{2}}\left(\sqrt{\alpha}\left(1-q^{2}\right)\Delta^{L-1}\left(q^{E_{11}}E_{10}\right)\right),$
(55)

where we have additionally used the relation
$q^{E_{00}}E_{10}=q^{-1}E_{10}q^{E_{00}}$ and that $q^{-1}$ only contributes a
constant.

The matrix entries of $\mathfrak{S}_{\alpha}$ and
$\hat{\mathfrak{S}}_{\alpha}$ are explicitly calculated in the unnumbered
equations before (99) in [CFG21], and produce the single–species duality
function. Therefore, for the value of $\lambda=\sqrt{\alpha}(1-q^{2})$, we
have proved the theorem in the single–species case.

####  4.5.2 Multi–species case

In the multi–species case, the parameter $\lambda$ in $U_{i}(\lambda)$ is
chosen so that $\lambda_{i}=\sqrt{\alpha_{i}}(1-q^{2})$. Then the duality
function is not simply a product of the single–species duality functions.
However, it is “almost” a product of single–species duality functions, in the
sense that it is immediate that

$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})=C_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})\prod_{i=0}^{n-1}D^{\boldsymbol{\theta}^{(i)}}_{\alpha_{i}}(\boldsymbol{\xi}_{i},\boldsymbol{\zeta}^{(i)}_{i})$

for some correction term $C_{\boldsymbol{\alpha}}$. This is immediate because
the unitary symmetry in the multi–species case is a product of the symmetries
in the single–species case, but with diagonal matrices in between terms of
this product. Thus, to complete the proof, it only remains to compute the term
$C_{\alpha}.$

From the general algebraic set–up in Theorem 4.5(ii), the duality functions
$\mathcal{D}^{\boldsymbol{\theta}}_{\boldsymbol{\alpha}}(\boldsymbol{\xi},\boldsymbol{\eta})$
are orthogonal with respect to the reversible measures defined by
$A^{2}G^{-2}B^{2}$. Given that part (b) of the Theorem has already been
proven, it therefore suffices to show that

$A^{2}G^{-2}B^{2}(\boldsymbol{\xi},\boldsymbol{\xi})=\mu^{n}(\boldsymbol{\xi}).$

But for the choice of the diagonal matrix with entries

$A(\boldsymbol{\xi},\boldsymbol{\xi})=\sum_{\boldsymbol{k}}a_{\boldsymbol{k}}\cdot
1_{\left\\{N\left(\xi_{i}\right)=k_{i},0\leq i\leq n\right\\}},$

this follows immediately from (40) and the definition of
$\mu^{n}(\boldsymbol{\xi})$.

###  4.6 Proof of Theorem 3.2: duality for the multi–species $q$–TAZRP

First, note that the two expressions in the theorem are equal by equation
(1.11.2) of [KLS], which is a $q-$analogue of Newton’s binomium

${}_{1}\phi_{0}\left(\begin{array}[]{c}q^{-n}\\\
-\end{array};,z\right)=\left(zq^{-n};q\right)_{n},\quad n=0,1,2,\ldots$

####  4.6.1 Charge–Parity Symmetry

If the limit $\boldsymbol{\theta}\rightarrow\infty$ is taken in the duality
function, the result is $0$. Indeed, in the totally asymmetric case, this must
always be the case, as explained in [Kua17], and also observed in [CGRS16] for
triangular self-duality functions. But with charge parity, the duality result
follows immediately if we can take the limit of the
ASEP$(q,\boldsymbol{\theta})$ duality function and show that it equals the
function in the theorem (up to constants under the dynamics), the result is
proven. The remainder of this subsubsection is to that calculation.

First we consider the single species ASEP$(q,\boldsymbol{\theta})$.

######  Theorem 4.6.

$\prod_{x\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-2\xi^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1}\right)$
(56)

is a space reversed duality for q-TAZRP, where $\boldsymbol{\eta}$ evolves
with total asymmetry to the left and $\boldsymbol{\xi}$ evolves with total
asymmetry to the right.

###### Proof.

Recall the orthogonality duality for ASEP:

$D_{\alpha}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})=\prod_{x=1}^{L}K_{\eta^{x}}\left(q^{-2\xi^{x}},p^{x}(\boldsymbol{\xi},\boldsymbol{\eta}),\theta^{x},q^{2}\right)$
(57)

and

$p^{x}(\boldsymbol{\xi},\boldsymbol{\eta})=\alpha^{-1}q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi})-N_{x+1}^{+}(\boldsymbol{\eta})\right)+2N_{x-1}^{-}(\boldsymbol{\theta})-1},$
(58)

where $\alpha$ satisfies $q^{2\theta^{x}}p^{x}(\xi,\eta)>1$ for all $x$, thus
$0<\alpha<q^{2N(\boldsymbol{\theta})-1}$.

Now apply $T$ to $\boldsymbol{\eta}$,

$\begin{array}[]{cl}D_{\alpha}^{\boldsymbol{\theta}}\left(\boldsymbol{\xi},T(\boldsymbol{\eta})\right)&=\prod_{x=1}^{L}K_{\theta^{x}-\eta^{x}}\left(q^{-2\xi^{x}},p^{x}(\boldsymbol{\xi},\boldsymbol{\theta}-\boldsymbol{\eta}),\theta^{x},q^{2}\right)\\\
&=\prod_{x=1}^{L}{}_{2}\phi_{1}(q^{-2\xi^{x}},q^{-2(\theta^{x}-\eta^{x})};q^{-2\theta^{x}};q^{2},p^{x}(\boldsymbol{\xi},\boldsymbol{\theta}-\boldsymbol{\eta})q^{2(\theta^{x}-\eta^{x}+1)}).\end{array}$
(59)

Take $\alpha=q^{2N(\boldsymbol{\theta})}$, Next we take $\boldsymbol{\theta}$
to $\infty$, i.e $\theta^{x}\xrightarrow[]{}\infty$ for all $x$,

$\begin{array}[]{l}\displaystyle{\lim_{\boldsymbol{\theta}\xrightarrow{}\infty}{}_{2}\phi_{1}(q^{-2\xi^{x}},q^{-2(\theta^{x}-\eta^{x})};q^{-2\theta^{x}};q^{2},p^{x}(\boldsymbol{\xi},\boldsymbol{\theta}-\boldsymbol{\eta})q^{2(\theta^{x}-\eta^{x}+1)})}\\\
=\displaystyle{\lim_{\boldsymbol{\theta}\xrightarrow{}\infty}{}_{2}\phi_{1}\left(q^{-2\xi^{x}},q^{-2(\theta^{x}-\eta^{x})};q^{-2\theta^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)-2\eta^{x}+1}\right)}\\\
=\displaystyle{\sum_{k=0}^{\xi^{x}}\frac{(q^{-2\xi^{x}};q^{2})_{k}}{(q^{2};q^{2})_{k}}\left(q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1}\right)^{k}}\\\
={}_{1}\phi_{0}\left(q^{-2\xi^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1}\right).\end{array}$
(60)

∎

Here is the generalized duality for general $n$-species $q$-TAZRP .

######  Theorem 4.7.

$D^{q-TARZP}(\boldsymbol{\xi},\boldsymbol{\eta})=q^{h(\boldsymbol{\xi},\boldsymbol{\eta})}\prod_{i=0}^{n-1}\prod_{x=1}^{L}{}_{1}\phi_{0}\left(q^{-2\xi_{i}^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1}\right)$
(61)

is a space reversed duality for $n$-species $q$-TAZRP , where
$\boldsymbol{\eta}$ evolves with total asymmetry to the left and
$\boldsymbol{\xi}$ evolves with total asymmetry to the right, and

$h(\boldsymbol{\xi},\boldsymbol{\eta})=\sum_{x\in\Lambda_{L}}\sum_{i=0}^{n-1}\left(-\xi_{i}^{x}N_{x+1}^{+}(\boldsymbol{\eta}_{[0,n-2-i]})+\eta_{i}^{x}N_{x}^{+}(\boldsymbol{\xi}_{[0,n-2-i]})\right).$
(62)

###### Proof.

Let
$\boldsymbol{\theta}^{(i)}=\boldsymbol{\eta}_{[0,i+1]}-\boldsymbol{\xi}_{[0,i-1]}$,
$\alpha_{i}=q^{2N(\boldsymbol{\theta}^{(i)})}$,

$\begin{split}\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},\boldsymbol{\eta})=\prod_{i=0}^{n-1}D_{\alpha_{i}}^{(\boldsymbol{\theta}^{(i)})}(\boldsymbol{\xi}_{i},\boldsymbol{\eta}_{[0,i]}-\boldsymbol{\xi}_{[0,i-1]})\times\sqrt{\frac{\prod_{0=1}^{n-1}\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\xi}_{i})\mu_{\alpha_{i}}^{\boldsymbol{\theta}^{(i)}}(\boldsymbol{\eta}_{[0,i]}-\boldsymbol{\xi}_{[0,i-1]})}{\mu^{n}(\boldsymbol{\xi})\mu^{n}(\boldsymbol{\eta})}},\end{split}$
(63)

We claim that there exists function
$f(\boldsymbol{\theta},N(\boldsymbol{\xi}),N(\boldsymbol{\eta}))$ and
$h(\boldsymbol{\xi},\boldsymbol{\eta})$ such that

$\begin{split}\lim_{\boldsymbol{\theta}\xrightarrow{}\infty}\frac{\mathcal{D}_{\boldsymbol{\alpha}}^{\boldsymbol{\theta}}(\boldsymbol{\xi},T(\boldsymbol{\eta}))}{q^{\frac{f(\boldsymbol{\theta},N(\boldsymbol{\xi}),N(\boldsymbol{\eta}))}{2}}}=q^{h(\boldsymbol{\xi},\boldsymbol{\eta})}\prod_{i=0}^{n-1}\prod_{x=1}^{L}{}_{1}\phi_{0}\left(q^{-2\xi_{i}^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1}\right).\end{split}$
(64)

Next, compute the limit of

$\prod_{x\in\Lambda_{L}}\frac{\prod_{i=0}^{n-1}\mu_{T(\alpha_{i})}^{T(\boldsymbol{\theta}^{(i)})}(\boldsymbol{\xi}_{i})\mu_{T(\alpha_{i})}^{T(\boldsymbol{\theta}^{(i)})}(T(\eta_{[0,i]})-\xi_{[0,i-1]})}{\mu^{n}(\boldsymbol{\xi})\mu^{n}(T(\boldsymbol{\eta}))},$
(65)

where
$T(\boldsymbol{\theta}^{(i)})=\boldsymbol{\theta}-\boldsymbol{\eta}_{[0,n-2-i]}-\boldsymbol{\xi}_{[0,i-1]}$,
$T(\boldsymbol{\eta}_{[0,i]})=\boldsymbol{\theta}-\boldsymbol{\eta}_{[0,n-1-i]}$.

Split this term into two parts, one contains q-binom terms and the other is
$q^{r(\boldsymbol{\theta},\boldsymbol{\xi},\boldsymbol{\eta})}$.

$\begin{split}\lim_{\boldsymbol{\theta}\xrightarrow{}\infty}\prod_{x\in\Lambda_{L}}\frac{\prod_{i=0}^{n-1}\binom{T(\theta^{(i)})^{x}}{\xi_{i}^{x}}_{q}\binom{T(\theta^{(i)})^{x}}{T(\eta_{[0,i]})^{x}-\xi^{x}_{[0,i-1]}}_{q}}{\binom{\theta^{x}}{\boldsymbol{\xi}^{x}}_{q}\binom{\theta^{x}}{\boldsymbol{\eta}^{x}}_{q}}=\lim_{\boldsymbol{\theta}\xrightarrow{}\infty}\prod_{x\in\Lambda_{L}}\left(\frac{\prod_{i=0}^{n-1}[\theta^{x}-\eta^{x}_{[0,n-2-i]}-\xi^{x}_{[0,i-1]}]^{!}_{q}}{[\theta^{x}]_{q}^{!}\prod_{i=0}^{n-2}[\theta^{x}-\eta^{x}_{[0,n-2-i]}-\xi^{x}_{[0,i]}]^{!}_{q}}\right)^{2}\\\
=q^{2\sum_{x}\sum_{i=0}^{n-1}\xi^{x}_{n-2-i}\eta^{x}_{[0,i]}},\end{split}$
(66)

while

$r(\boldsymbol{\theta},\boldsymbol{\xi},\boldsymbol{\eta})=\sum_{i=0}^{n-1}\sum_{x\in\Lambda_{L}}(2N_{x+1}^{+}(T(\boldsymbol{\theta}^{(i)}))+T(\theta^{(i)})^{x})(T(\eta_{[0,i]})^{x}-\xi_{[0,i-1]}^{x}+\xi_{i}^{x})\\\
-\sum_{x\in\Lambda_{L}}\sum_{i=0}^{n}\frac{(\xi_{i}^{x})^{2}+(\eta_{i}^{x})^{2}}{2}+\sum_{x\in\Lambda_{L}}\sum_{y<x}\sum_{i=0}^{n-1}2\xi^{x}_{[0,i]}\xi_{i+1}^{y}+\sum_{x\in\Lambda_{L}}\sum_{y<x}\sum_{i=0}^{n-1}2T(\eta)^{x}_{[0,i]}T(\eta)_{i+1}^{y}\\\
-2\sum_{i=0}^{n-1}N(T(\boldsymbol{\theta}^{(i)}))N(\boldsymbol{\xi}_{i})-2\sum_{i=0}^{n-1}N(T(\boldsymbol{\theta}^{(i)}))N(T(\boldsymbol{\eta}_{[0,i]})-\boldsymbol{\xi}_{[0,i-1]})).$
(67)

We claim that $r(\boldsymbol{\theta},\boldsymbol{\xi},\boldsymbol{\eta})$
could be written as a sum of
$f(\boldsymbol{\theta},N(\boldsymbol{\xi}),N(\boldsymbol{\eta}))$ and a
function $g(\boldsymbol{\xi},\boldsymbol{\eta})$ that does not depend on
$\boldsymbol{\theta}$, we can divide the duality function by $q^{f/2}$, thus
get a nontrivial limit.

$r(\boldsymbol{\theta},\xi,\eta)=f(\boldsymbol{\theta},N(\boldsymbol{\xi}),N(\boldsymbol{\eta}))+g(\boldsymbol{\xi},\boldsymbol{\eta}),$
(68)

where

$\begin{split}g(\boldsymbol{\xi},\boldsymbol{\eta})=2\sum_{x\in\Lambda_{L}}\sum_{i=0}^{n-1}\left(-\xi_{i}^{x}N_{x+1}^{+}(\boldsymbol{\eta}_{[0,n-2-i]})+\eta_{i}^{x}N_{x+1}^{+}(\boldsymbol{\xi}_{[0,n-2-i]})\right).\end{split}$
(69)

Note that when $n=1$, $g=0$, so it degenerates to the single-species case as
well.

∎

###  4.7 Proof of Theorem 3.3: duality for the stochastic multi–species
higher–spin vertex model

Although this proof is difficult from a probabilistic perspective, it is
straightforward from an algebraic perspective. Because the argument is exactly
the same as in [Kua18a], we only outline the idea. The transition matrix for
the vertex model is known as the stochastic transfer matrix, and is a product
of the matrices $S(z)$ from section 2.6. These matrices satisfy

$S(z)\Delta(u)=\Delta(u)S^{\text{rev}}(z)$

for any $u\in\mathcal{U}_{q}(\mathfrak{gl}_{n+1})$, where

$S^{\text{rev}}(z)=P\circ S(z)$

with $P$ being the permutation operator

$P(v\otimes w)=w\otimes v.$

The permutation operator can be interpreted probabilistically as the
charge–parity symmetry. The key difference between this paper and [Kua18a] is
a different choice of $u$, which does not change the fact that resulting
function is a duality function.

###  4.8 Proof of Theorem 3.4: duality for the multi–species $q$-Hahn TAZRP

First, a few identities will be needed. These will be stated as the below
lemmas:

######  Lemma 4.8.

Fix $|q|<1$ and $0\leq\mu<1,0\leq\lambda\leq 1$. Then for all non–negative
integers $x$ and $y$, and any real number $c$,

$\sum_{j=0}^{x}\Phi_{q}(j\mid
x,\lambda,\mu)\frac{\left(q^{-c+j}\right)_{y}}{\left(q^{-c}\right)_{y}}=\sum_{s=0}^{y}\Phi_{q}(s\mid
y,\lambda,\mu)\frac{\left(q^{-c+s}\right)_{x}}{\left(q^{-c}\right)_{x}}.$

###### Proof.

Let
$S_{x,y}=\sum_{j=0}^{x}\Phi_{q}(j|x,\lambda,\mu)\frac{(q^{-c+j})_{y}}{(q^{-c})_{y}}$.
First note that

$S_{x,y}=\frac{(\mu/\lambda)_{x}}{(\mu)_{x}}{}_{3}\phi_{2}\left(\begin{matrix}\lambda&q^{-x}&q^{-c+y}\\\
\frac{\lambda}{\mu}q^{1-x}&&q^{-c}\end{matrix};q,q\right).$ (70)

Recall a transformation formula for ${}_{3}\phi_{2}$ [Gas96]:

${}_{3}\phi_{2}\left(\begin{matrix}q^{-n}&a&b\\\
d&&e\end{matrix};q,q\right)=\frac{(a)_{n}(e/b)_{n}}{(d)_{n}(e)_{n}}b^{n}{}_{3}\phi_{2}\left(\begin{matrix}q^{-n}&d/a&q^{1-n}/e\\\
q^{1-n}/a&&q^{1-n}b/d\end{matrix};q,q\right).$ (71)

Let $n=x$, $a=q^{-c+y}$, $b=\lambda$, $d=q^{-c}$,
$e=\frac{\lambda}{\mu}q^{1-x}$,

$\displaystyle S_{x,y}$
$\displaystyle=\frac{\left(\mu/\lambda\right)_{x}\left(q^{-c+y}\right)_{x}\left(\frac{1}{\mu}q^{1-x}\right)_{x}}{\left(\mu\right)_{x}\left(q^{-c}\right)_{x}\left(\frac{\lambda}{\mu}q^{1-x}\right)_{x}}\lambda^{x}{}_{3}\phi_{2}\left(\begin{matrix}q^{-x}&q^{-y}&\frac{\mu}{\lambda}\\\
q^{1-x-y+c}&&\mu\end{matrix};q,q\right)$
$\displaystyle=\frac{\left(q^{-c+y}\right)_{x}}{\left(q^{-c}\right)_{x}}{}_{3}\phi_{2}\left(\begin{matrix}q^{-x}&q^{-y}&\frac{\mu}{\lambda}\\\
q^{1-x-y+c}&&\mu\end{matrix};q,q\right)$
$\displaystyle=\frac{\left(q^{-c+x}\right)_{y}}{\left(q^{-c}\right)_{y}}{}_{3}\phi_{2}\left(\begin{matrix}q^{-x}&q^{-y}&\frac{\mu}{\lambda}\\\
q^{1-x-y+c}&&\mu\end{matrix};q,q\right)$ $\displaystyle=S_{y,x}.$

∎

Take derivative of lemma 4.8 w.r.t. $\lambda$ and let $\lambda=1$ yields the
following identity.

######  Corollary 4.9.

$\sum_{0\leq j\leq
x}\Phi^{\prime}(j|x,\mu)\frac{(q^{-c+y})_{j}}{(q^{-c})_{j}}=\sum_{0\leq s\leq
y}\Phi^{\prime}(s|y,\mu)\frac{(q^{-c+x})_{s}}{(q^{-c})_{s}}$ (72)

for $x,y,\in\mathbb{N},c\in\mathbb{R}$.

In what follows, $\xi_{[n_{1},n_{2}]}=\sum_{i=n_{1}}^{n_{2}}\xi_{i}$, if
$n_{1}>n_{2}$, then $\xi_{[n_{1},n_{2}]}=0$.
$\boldsymbol{\xi}_{[n_{1},n_{2}]}=(\xi_{n_{1}},\ldots,\xi_{n_{2}})$.

Here is a multi-species generalization:

######  Proposition 4.10.

(1)Fix $|q|<1$ and $0\leq\mu<1$, $0\leq\lambda\leq 1$, then for all
$\boldsymbol{\eta},\boldsymbol{\xi}\in\mathbb{Z}^{n}_{\geq 0}$, and
$\boldsymbol{c}\in\mathbb{R}^{n}$.

$\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\eta}}\Phi(\boldsymbol{\gamma}|\boldsymbol{\eta},\lambda,\mu)\prod_{i=0}^{n-1}q^{\boldsymbol{\gamma}_{i}\xi_{[0,n-2-i]}}\frac{\left(q^{-c_{i}+\gamma_{n-1-i}}\right)_{\xi_{i}}}{\left(q^{-c_{i}}\right)_{\xi_{i}}}=\sum_{\boldsymbol{\zeta}\leq\boldsymbol{\xi}}\Phi(\boldsymbol{\zeta}|\boldsymbol{\xi},\lambda,\mu)\prod_{i=0}^{n-1}q^{\zeta_{i}\eta_{[0,n-2-i]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-1-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-1-i}}}.$
(73)

(2)Fix $|q|<1$ and $0\leq\mu<1$, then for all
$\boldsymbol{\eta},\boldsymbol{\xi}\in\mathbb{Z}^{n}_{\geq 0}$, and
$\boldsymbol{c}\in\mathbb{R}^{n}$

$\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\eta}}\Phi^{\prime}(\boldsymbol{\gamma}|\boldsymbol{\eta},\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-2-i]}}\frac{\left(q^{-c_{i}+\gamma_{n-1-i}}\right)_{\xi_{i}}}{\left(q^{-c_{i}}\right)_{\xi_{i}}}=\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\xi}}\Phi^{\prime}(\boldsymbol{\zeta}|\boldsymbol{\xi},\mu)\prod_{i=0}^{n-1}q^{\zeta_{i}\eta_{[0,n-2-i]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-1-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-1-i}}}.$
(74)

###### Proof.

Induction on $n$, the base case when $n=1$ is just Lemma 4.8, now assume (73)
holds for $n$-species, next we show the identity holds also for
$(n+1)$-species.

Now suppose $\boldsymbol{\eta},\boldsymbol{\xi}\in\mathbb{Z}^{n+1}_{\geq 0}$,
and $\boldsymbol{c}\in\mathbb{R}^{n+1}$,

$\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\eta}}\Phi(\boldsymbol{\gamma}|\boldsymbol{\eta},\lambda,\mu)\prod_{i=0}^{n}q^{\gamma_{i}\xi_{[0,n-1-i]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}=\\\
\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\lambda,\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-1-i]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}\\\
\sum_{\gamma_{n}\leq\eta_{n}}\left(\frac{\mu}{\lambda}\right)^{\gamma_{n}}q^{\gamma_{n}(\eta_{[0,n-1]}-\gamma_{[0,n-1]})}\frac{(\lambda
q^{\gamma_{[0,n-1]}})_{\gamma_{n}}(\frac{\mu}{\lambda}q^{\eta_{[0,n-1]}-\gamma_{[0,n-1]}})_{\eta_{n}-\gamma_{n}}}{(\mu
q^{\eta_{[0,n-1]}})_{\eta_{n}}}\binom{\eta_{n}}{\gamma_{n}}_{q}\frac{\left(q^{-c_{0}+\gamma_{n}}\right)_{\xi_{0}}}{\left(q^{-c_{0}}\right)_{\xi_{0}}}.$
(75)

Let $\hat{\lambda}=\lambda q^{\gamma_{[0,n-1]}}$, $\hat{\mu}=\mu
q^{\eta_{[0,n-1]}}$, by Lemma 4.8,

$\begin{split}\sum_{\gamma_{n}\leq\eta_{n}}\left(\frac{\mu}{\lambda}\right)^{\gamma_{n}}q^{\gamma_{n}(\eta_{[0,n-1]}-\gamma_{[0,n-1]})}\frac{(\lambda
q^{\gamma_{[0,n-1]}})_{\gamma_{n}}(\frac{\mu}{\lambda}q^{\eta_{[0,n-1]}-\gamma_{[0,n-1]}})_{\eta_{n}-\gamma_{n}}}{(\mu
q^{\eta_{[0,n-1]}})_{\eta_{n}}}\binom{\eta_{n}}{\gamma_{n}}_{q}\frac{\left(q^{-c_{0}+\gamma_{n}}\right)_{\xi_{0}}}{\left(q^{-c_{0}}\right)_{\xi_{0}}}\\\
=\sum_{\gamma_{n}\leq\eta_{n}}\Phi(\gamma_{n}|\eta_{n},\hat{\lambda},\hat{\mu})\frac{\left(q^{-c_{0}+\gamma_{n}}\right)_{\xi_{0}}}{\left(q^{-c_{0}}\right)_{\xi_{0}}}=\sum_{\zeta_{0}\leq\xi_{0}}\Phi(\zeta_{0}|\xi_{0},\hat{\lambda},\hat{\mu})\frac{\left(q^{-c_{0}+\zeta_{0}}\right)_{\eta_{n}}}{\left(q^{-c_{0}}\right)_{\eta_{n}}}.\end{split}$
(76)

Let $\tilde{\lambda}=\lambda q^{\zeta_{0}}$, $\tilde{\mu}=\mu q^{\xi_{0}}$,

$\displaystyle\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\lambda,\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-i-1]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}$
$\displaystyle\quad\quad\quad\times
q^{\zeta_{0}(\eta_{[0,n-1]}-\gamma_{[0,n-1]})}\frac{(\lambda
q^{\gamma_{[0,n-1]}})_{\zeta_{0}}(\frac{\mu}{\lambda}q^{\eta_{[0,n-1]}-\gamma_{[0,n-1]}})_{\xi_{0}-\zeta_{0}}}{(\mu
q^{\eta_{[0,n-1]}})_{\xi_{0}}}$ $\displaystyle=$ $\displaystyle
q^{\zeta_{0}\eta_{[0,n-1]}}\frac{(\lambda)_{\zeta_{0}}(\frac{\mu}{\lambda})_{\xi_{0}-\zeta_{0}}}{(\mu)_{\xi_{0}}}\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\tilde{\lambda},\tilde{\mu})\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[1,n-i-1]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}$
$\displaystyle=$ $\displaystyle
q^{\zeta_{0}\eta_{[0,n-1]}}\frac{(\lambda)_{\zeta_{0}}(\frac{\mu}{\lambda})_{\xi_{0}-\zeta_{0}}}{(\mu)_{\xi_{0}}}\sum_{\boldsymbol{\zeta}_{[1,n]}\leq\boldsymbol{\xi}_{[1,n]}}\Phi(\boldsymbol{\zeta}_{[1,n]}|\boldsymbol{\xi}_{[1,n]},\tilde{\lambda},\tilde{\mu})\prod_{i=0}^{n-1}q^{\zeta_{i+1}\eta_{[0,n-i-2]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-i}}}$
$\displaystyle=$ $\displaystyle
q^{\zeta_{0}\eta_{[0,n-1]}}\frac{(\lambda)_{\zeta_{0}}(\frac{\mu}{\lambda})_{\xi_{0}-\zeta_{0}}}{(\mu)_{\xi_{0}}}\sum_{\boldsymbol{\zeta}_{[1,n]}\leq\boldsymbol{\xi}_{[1,n]}}\Phi(\boldsymbol{\zeta}_{[1,n]}|\boldsymbol{\xi}_{[1,n]},\tilde{\lambda},\tilde{\mu})\prod_{i=1}^{n}q^{\zeta_{i}\eta_{[0,n-i-1]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-i}}},$

where the second to last equality follows from the induction hypothesis. Thus
plug in the above two equations,

$\text{\eqref{eq:3}}=\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\lambda,\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-1-i]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}\sum_{\zeta_{0}\leq\xi_{0}}\Phi(\zeta_{0}|\xi_{0},\hat{\lambda},\hat{\mu})\frac{\left(q^{-c_{0}+\zeta_{0}}\right)_{\eta_{n}}}{\left(q^{-c_{0}}\right)_{\eta_{n}}}\\\
=\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\lambda,\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-1-i]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}\\\
\sum_{\zeta_{0}\leq\xi_{0}}\left(\frac{\mu}{\lambda}\right)^{\zeta_{0}}q^{\zeta_{0}(\eta_{[0,n-1]}-\gamma_{[0,n-1]})}\frac{(\lambda
q^{\gamma_{[0,n-1]}})_{\zeta_{0}}(\frac{\mu}{\lambda}q^{\eta_{[0,n-1]}-\gamma_{[0,n-1]}})_{\xi_{0}-\zeta_{0}}}{(\mu
q^{\eta_{[0,n-1]}})_{\xi_{0}}}\binom{\xi_{0}}{\zeta_{0}}_{q}\frac{\left(q^{-c_{0}+\zeta_{0}}\right)_{\eta_{n}}}{\left(q^{-c_{0}}\right)_{\eta_{n}}}\\\
=\sum_{\zeta_{0}\leq\xi_{0}}\left(\frac{\mu}{\lambda}\right)^{\zeta_{0}}\binom{\xi_{0}}{\zeta_{0}}_{q}\frac{\left(q^{-c_{0}+\zeta_{0}}\right)_{\eta_{n}}}{\left(q^{-c_{0}}\right)_{\eta_{n}}}\sum_{\boldsymbol{\gamma}_{[0,n-1]}\leq\boldsymbol{\eta}_{[0,n-1]}}\Phi(\boldsymbol{\gamma}_{[0,n-1]}|\boldsymbol{\eta}_{[0,n-1]},\lambda,\mu)\prod_{i=0}^{n-1}q^{\gamma_{i}\xi_{[0,n-1-i]}}\frac{\left(q^{-c_{n-i}+\gamma_{i}}\right)_{\xi_{n-i}}}{\left(q^{-c_{n-i}}\right)_{\xi_{n-i}}}\\\
\times q^{\zeta_{0}(\eta_{[0,n-1]}-\gamma_{[0,n-1]})}\frac{(\lambda
q^{\gamma_{[0,n-1]}})_{\zeta_{0}}(\frac{\mu}{\lambda}q^{\eta_{[0,n-1]}-\gamma_{[0,n-1]}})_{\xi_{0}-\zeta_{0}}}{(\mu
q^{\eta_{[0,n-1]}})_{\xi_{0}}}\\\
=\sum_{\zeta_{0}\leq\xi_{0}}\left(\frac{\mu}{\lambda}\right)^{\zeta_{0}}\binom{\xi_{0}}{\zeta_{0}}_{q}\frac{\left(q^{-c_{0}+\zeta_{0}}\right)_{\eta_{n}}}{\left(q^{-c_{0}}\right)_{\eta_{n}}}q^{\zeta_{0}\eta_{[0,n-1]}}\frac{(\lambda)_{\zeta_{0}}(\frac{\mu}{\lambda})_{\xi_{0}-\zeta_{0}}}{(\mu)_{\xi_{0}}}\\\
\sum_{\boldsymbol{\zeta}_{[1,n]}\leq\boldsymbol{\xi}_{[1,n]}}\Phi(\boldsymbol{\zeta}_{[1,n]}|\boldsymbol{\xi}_{[1,n]},\lambda
q^{\zeta_{0}},\mu
q^{\xi_{0}})\prod_{i=1}^{n}q^{\zeta_{i}\eta_{[0,n-1-i]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-i}}}\\\
=\sum_{\boldsymbol{\zeta}\leq\boldsymbol{\xi}}\Phi(\boldsymbol{\zeta}|\boldsymbol{\xi},\lambda,\mu)\prod_{i=0}^{n}q^{\zeta_{i}\eta_{[0,n-1-i]}}\frac{\left(q^{-c_{i}+\zeta_{i}}\right)_{\eta_{n-i}}}{\left(q^{-c_{i}}\right)_{\eta_{n-i}}}.$
(77)

The second part of the proposition is the derivative of the first part with
respect to $\lambda$ at $\lambda=1$.

∎

With the identities proven, we now turn to the conjecture that the duality
function obtained in previous section is a space–reversed self–duality
function for the single–species $q^{2}$-Hahn Boson process.

######  Theorem 4.11.

$\displaystyle
D(\boldsymbol{\eta},\boldsymbol{\xi})=\prod_{x\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-\xi^{x}};q,q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)$
(78)
$\displaystyle=\prod_{x\in\Lambda_{L}}\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}\
\ \ \ \ \ \ \ \ \ \ \ \ \ $

is a space reversed self-duality function for $q$-Hahn Boson process where
$\eta$ evolves with total asymmetry to the left and $\xi$ evolves with total
asymmetry to the right.

###### Proof.

Let

$D_{x}(\boldsymbol{\eta},\boldsymbol{\xi})=\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}.$
(79)

Recall that because the process is a nearest neighbour zero–range process the
generator of $\boldsymbol{\eta}$ can be written as
$\mathcal{L}=\sum_{x=1}^{L-1}\mathcal{L}_{x+1}$, generator of
$\boldsymbol{\xi}$ can be written as
$\tilde{\mathcal{L}}=\sum_{x=1}^{L-1}\tilde{\mathcal{L}}_{x}$, where
$\mathcal{L}_{x}$ are the two–site generators.

Since $D_{x}(\boldsymbol{\eta},\boldsymbol{\xi})$ involves counting the number
of particles in $\boldsymbol{\eta}$ at sites to the right of $x$ (exclusive),
if $\mathcal{L}(\boldsymbol{\eta},\boldsymbol{\sigma})\neq 0$ then
$D_{y}(\boldsymbol{\sigma},\boldsymbol{\xi})=D_{y}(\boldsymbol{\eta},\boldsymbol{\xi})$
for $y\neq x$. Let
$\boldsymbol{\sigma}=\boldsymbol{\eta}+\gamma^{(x)}-\gamma^{(x+1)}$ for some
$\gamma\in\mathbb{Z}$, where $\gamma^{(x)}$ denotes $\gamma$ particles at site
$x$, (i.e. $\boldsymbol{\sigma}$ is obtained from $\eta$ by moving $\gamma$
particles from $x$ to $x+1$)

$D(\boldsymbol{\sigma},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+\gamma+1/2}\right)_{\xi^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}}.$
(80)

Let
$\tilde{\boldsymbol{\sigma}}=\boldsymbol{\xi}+\gamma^{(x+1)}-\gamma^{(x)}$,

$\begin{array}[]{cl}D(\boldsymbol{\eta},\tilde{\boldsymbol{\sigma}})&=D(\boldsymbol{\eta},\boldsymbol{\xi})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma}\right)_{\xi^{x}-\gamma}\left(q^{-\left(N_{x+1}^{-}(\boldsymbol{\xi})+N_{x+2}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x+1}+\gamma}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}\left(q^{-\left(N_{x+1}^{-}(\boldsymbol{\xi})+N_{x+2}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x+1}}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+2}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\gamma}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\gamma}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma}\right)_{\eta^{x+1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x+1}}}.\end{array}$
(81)

$\mathcal{L}D(\boldsymbol{\eta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{x=1}^{L-1}\sum_{\gamma}\mathcal{L}_{x+1}(\boldsymbol{\eta},\boldsymbol{\eta}+\gamma^{(x)}-\gamma^{(x+1)})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+\gamma+1/2}\right)_{\xi^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}}$
(82)

$D\tilde{\mathcal{L}}^{*}(\boldsymbol{\eta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{x=1}^{L-1}\sum_{\gamma}\mathcal{\tilde{L}}_{x}(\boldsymbol{\xi},\boldsymbol{\xi}+\gamma^{(x+1)}-\gamma^{(x)})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma}\right)_{\eta^{x+1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x+1}}}$
(83)

Therefore, it suffices to show

$\sum_{0\leq\gamma\leq\eta^{x+1}}\Phi^{\prime}(\gamma|\eta^{x+1})\frac{\left(q^{-\left(N_{x}^{-}(\xi)+N_{x+1}^{+}(\eta)\right)+\gamma+1/2}\right)_{\xi^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}}=\sum_{0\leq\gamma\leq\xi^{x}}\Phi^{\prime}(\gamma|\xi^{x})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma}\right)_{\eta^{x+1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x+1}}}.$
(84)

This is just Corollary 4.9.

Next turn to the discrete-time $q$-Hahn Boson process. If the evolution is to
the left, then the transition probabilities are

$P(\boldsymbol{\eta},\boldsymbol{\zeta})=\prod_{x=2}^{L}\Phi(\gamma^{x}|\eta^{x}),$
(85)

where $\zeta^{x}=\eta^{x}-\gamma^{x}+\gamma^{x+1}$, $1\leq x\leq L$ and
$\gamma^{1}=\gamma^{L+1}=0$. In words, this denotes that $\gamma^{x}$
particles have left lattice site $x$ in particle configuration
$\boldsymbol{\eta}$. For each $x$,

$D_{x}(\boldsymbol{\zeta},\boldsymbol{\xi})=D_{x}(\boldsymbol{\eta},\boldsymbol{\xi})\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x+1}}\right)_{\xi^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}}.$
(86)

Let $\mathfrak{X}$ denote the set of particle configurations at one site and
$\mathfrak{X}^{L}$ the Cartesian product. Thus, after re-indexing,

$\begin{split}\sum_{\boldsymbol{\zeta}\in\mathfrak{X}^{L}}P(\boldsymbol{\eta},\boldsymbol{\zeta})D(\boldsymbol{\zeta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{\boldsymbol{\gamma}\in\\{0\\}\times\mathfrak{X}^{L-1}}\prod_{x=2}^{L}\Phi(\gamma^{x}|\eta^{x})\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x}}\right)_{\xi^{x-1}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x-1}}}.\end{split}$
(87)

If the evolution is to the right,

$P(\boldsymbol{\xi},\boldsymbol{\zeta})=\prod_{x=1}^{L-1}\Phi(\gamma^{x}|\xi^{x}),$
(88)

where $\zeta^{x}=\xi^{x}-\gamma^{x}+\gamma^{x-1}$, $1\leq x\leq L$ and
$\gamma^{0}=\gamma^{L}=0$. For each $x$,

$\begin{array}[]{cl}D(\boldsymbol{\eta},\boldsymbol{\zeta})&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x}}\right)_{\xi^{x}-\gamma^{x}+\gamma^{x-1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x}}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\xi^{x}}\right)_{\gamma^{x-1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\gamma^{x}}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+2}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\gamma^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\gamma^{x}}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x}}\right)_{\eta^{x+1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi})+N_{x+1}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x+1}}}.\end{array}$
(89)

Thus,

$\sum_{\boldsymbol{\zeta}\in\mathfrak{X}^{L}}D(\boldsymbol{\eta},\boldsymbol{\zeta})P(\boldsymbol{\xi},\boldsymbol{\zeta})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{\gamma\in\mathfrak{X}^{L-1}\times\\{0\\}}\prod_{x=2}^{L}\Phi(\gamma^{x-1}|\xi^{x-1})\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x-1}}\right)_{\eta^{x}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x}}}.$
(90)

Therefore, it suffices to show that

$\sum_{\gamma^{x}\in\mathfrak{X}}\Phi(\gamma^{x}|\eta^{x})\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x}}\right)_{\xi^{x-1}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\xi^{x-1}}}=\sum_{\gamma^{x-1}\in\mathfrak{X}}\Phi(\gamma^{x-1}|\xi^{x-1})\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2+\gamma^{x-1}}\right)_{\eta^{x}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi})+N_{x}^{+}(\boldsymbol{\eta})\right)+1/2}\right)_{\eta^{x}}}.$
(91)

This is just Lemma 4.8. ∎

Now for the multi–species case.

######  Theorem 4.12.

Recall the function $h$ defined in (62):

$h(\boldsymbol{\xi},\boldsymbol{\eta})=\sum_{x\in\Lambda_{L}}\sum_{i=0}^{n-1}\left(-\xi_{i}^{x}N_{x+1}^{+}(\boldsymbol{\eta}_{[0,n-2-i]})+\eta_{i}^{x}N_{x}^{+}(\boldsymbol{\xi}_{[0,n-2-i]})\right).$
(92)

Then

$\begin{array}[]{cl}D(\boldsymbol{\eta},\boldsymbol{\xi})&=q^{\frac{h(\boldsymbol{\xi},\boldsymbol{\eta})}{2}}\prod_{i=0}^{n-1}\prod_{x=1}^{L}{}_{1}\phi_{0}\left(q^{-\xi_{i}^{x}};q,q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1/2}\right)\\\
&=q^{\frac{h(\boldsymbol{\xi},\boldsymbol{\eta})}{2}}\prod_{i=0}^{n-1}\prod_{x=1}^{L}\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(T(\boldsymbol{\eta})_{i+1})\right)+1/2};q^{2}\right)_{\xi_{i}^{x}}\\\
&=q^{\frac{h(\boldsymbol{\xi},\boldsymbol{\eta})}{2}}\prod_{i=0}^{n-1}\prod_{x=1}^{L}\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2};q^{2}\right)_{\xi_{i}^{x}}\end{array}$
(93)

is a space reversed self-duality function for $n$-species $q$-Hahn Boson
process where $\boldsymbol{\eta}$ evolves with total asymmetry to the left and
$\boldsymbol{\xi}$ evolves with total asymmetry to the right.

###### Proof.

Recall that the generator of $\boldsymbol{\eta}$ can be written as
$\mathcal{L}=\sum_{x=1}^{L-1}\mathcal{L}_{x+1}$, generator of
$\boldsymbol{\xi}$ can be written as
$\tilde{\mathcal{L}}=\sum_{x=1}^{L-1}\tilde{\mathcal{L}}_{x}$.

Let
$\boldsymbol{\sigma}=\boldsymbol{\eta}+\boldsymbol{\gamma}^{(x)}-\boldsymbol{\gamma}^{(x+1)}$
for some $\boldsymbol{\gamma}\in\mathbb{Z}^{n}$,

$D(\boldsymbol{\sigma},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})q^{\sum_{i=0}^{n-1}\gamma_{i}\xi_{[0,n-2-i]}^{x}}\prod_{i=0}^{n-1}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+\gamma_{n-1-i}+1/2}\right)_{\xi_{i}^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x}}}.$
(94)

Let
$\tilde{\boldsymbol{\sigma}}=\boldsymbol{\xi}+\boldsymbol{\gamma}^{(x+1)}-\boldsymbol{\gamma}^{(x)}$,

$\begin{split}D(\boldsymbol{\eta},\tilde{\boldsymbol{\sigma}})=D(\boldsymbol{\eta},\boldsymbol{\xi})q^{\sum_{i=1}^{n-1}\gamma_{i}|\eta_{[0,n-2-i]}^{x+1}|}\prod_{i=0}^{n-1}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+2}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\gamma_{i}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\gamma_{i}}}.\\\
\end{split}$ (95)

$\mathcal{L}D(\boldsymbol{\eta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{x=1}^{L-1}\sum_{\boldsymbol{\gamma}}\mathcal{L}_{x+1}(\boldsymbol{\eta},\boldsymbol{\eta}+\boldsymbol{\gamma}^{(x)}-\boldsymbol{\gamma}^{(x+1)})\frac{D(\boldsymbol{\eta}+\boldsymbol{\gamma}^{(x)}-\boldsymbol{\gamma}^{(x+1)},\boldsymbol{\xi})}{D(\boldsymbol{\eta},\boldsymbol{\xi})}.$
(96)

$D\tilde{\mathcal{L}}^{*}(\boldsymbol{\eta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{x=1}^{L-1}\sum_{\boldsymbol{\gamma}}\mathcal{\tilde{L}}_{x}(\boldsymbol{\xi},\boldsymbol{\xi}+\boldsymbol{\gamma}^{(x+1)}-\boldsymbol{\gamma}^{(x)})\frac{D(\boldsymbol{\eta},\boldsymbol{\xi}+\boldsymbol{\gamma}^{(x+1)}-\boldsymbol{\gamma}^{(x)})}{D(\boldsymbol{\eta},\boldsymbol{\xi})}.$
(97)

Therefore, it suffices to show

$\begin{split}\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\eta}^{x+1}}\Phi^{\prime}(\boldsymbol{\gamma}|\boldsymbol{\eta}^{x+1})q^{\sum_{i=0}^{n-1}\gamma_{i}\xi_{[0,n-2-i]}^{x}}\prod_{i=0}^{n-1}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+\boldsymbol{\gamma}_{n-1-i}+1/2}\right)_{\xi_{i}^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x}}}\\\
=\sum_{\boldsymbol{\gamma}\leq\boldsymbol{\xi}^{x}}\Phi^{\prime}(\boldsymbol{\gamma}|\boldsymbol{\xi}^{x})q^{\sum_{i=0}^{n-1}\boldsymbol{\gamma}_{i}\eta_{[0,n-2-i]}^{x+1}}\prod_{i=0}^{n-1}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+2}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\gamma_{i}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\gamma_{i}}},\end{split}$
(98)

which is just Proposition 4.10.

Next turn to the discrete-time $q$-Hahn Boson process. If the evolution is to
the left, then the transition probabilities are

$P(\boldsymbol{\eta},\boldsymbol{\zeta})=\prod_{x=2}^{L}\Phi(\boldsymbol{\gamma}^{x}|\boldsymbol{\eta}^{x}),$
(99)

where
$\boldsymbol{\zeta}^{x}=\boldsymbol{\eta}^{x}-\boldsymbol{\gamma}^{x}+\boldsymbol{\gamma}^{x+1}$,
$1\leq x\leq L$ and
$\boldsymbol{\gamma}^{1}=\boldsymbol{\gamma}^{L+1}=(0,\ldots,0)$.

$D(\boldsymbol{\zeta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\prod_{i=0}^{n-1}q^{\gamma_{i}^{x+1}\xi_{[0,n-2-i]}^{x}}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{n-1-i}^{x+1}}\right)_{\xi_{i}^{x}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x}}}.$
(100)

Thus, after re-indexing,

$\sum_{\boldsymbol{\zeta}\in\mathfrak{X}^{L}}P(\boldsymbol{\eta},\boldsymbol{\zeta})D(\boldsymbol{\zeta},\boldsymbol{\xi})=D(\boldsymbol{\eta},\boldsymbol{\xi})\\\
\sum_{\boldsymbol{\gamma}\in\\{0\\}\times\mathfrak{X}^{L-1}}\prod_{x=2}^{L}\Phi(\boldsymbol{\gamma}^{x}|\boldsymbol{\eta}^{x})\prod_{i=0}^{n-1}q^{\gamma_{i}^{x}\xi_{[0,n-2-i]}^{x-1}}\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{n-1-i}^{x}}\right)_{\xi_{i}^{x-1}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x-1}}}.$
(101)

If the evolution is to the right,

$P(\boldsymbol{\xi},\boldsymbol{\zeta})=\prod_{x=1}^{L-1}\Phi(\boldsymbol{\gamma}^{x}|\boldsymbol{\xi}^{x}),$
(102)

where
$\boldsymbol{\zeta}^{x}=\boldsymbol{\xi}^{x}-\boldsymbol{\gamma}^{x}+\boldsymbol{\gamma}^{x-1}$,
$1\leq x\leq L$ and $\gamma^{0}=\gamma^{L}=(0,\ldots,0)$.

$\begin{array}[]{cl}D(\boldsymbol{\eta},\boldsymbol{\zeta})&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\prod_{i=0}^{n-1}q^{\gamma_{i}^{x-1}\eta_{[0,n-2-i]}^{x}}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma^{x}_{i}}\right)_{\xi_{i}^{x}-\gamma_{i}^{x}+\gamma_{i}^{x-1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x}}}\\\
&=D(\boldsymbol{\eta},\boldsymbol{\xi})\prod_{x=1}^{L}\prod_{i=0}^{n-1}q^{\gamma_{i}^{x-1}\eta_{[0,n-2-i]}^{x}}\frac{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{i}^{x}}\right)_{\eta_{n-1-i}^{x+1}}}{\left(q^{-\left(N_{x}^{-}(\boldsymbol{\xi}_{i})+N_{x+1}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\eta_{n-1-i}^{x+1}}}.\end{array}$
(103)

Thus,

$\sum_{\boldsymbol{\zeta}\in\mathfrak{X}^{L}}D(\boldsymbol{\eta},\boldsymbol{\zeta})P(\boldsymbol{\xi},\boldsymbol{\zeta})=D(\boldsymbol{\eta},\boldsymbol{\xi})\sum_{\boldsymbol{\gamma}\in\mathfrak{X}^{L-1}\times\\{0\\}}\prod_{x=2}^{L}\\\
\Phi(\boldsymbol{\gamma}^{x-1}|\boldsymbol{\xi}^{x-1})\prod_{i=0}^{n-1}q^{\gamma_{i}^{x-1}\eta_{[0,n-2-i]}^{x}}\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{i}^{x-1}}\right)_{\eta_{n-1-i}^{x}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\eta_{n-1-i}^{x}}}.$
(104)

Therefore, it suffices to show that

$\sum_{\boldsymbol{\gamma}^{x}\in\mathfrak{X}}\Phi(\boldsymbol{\gamma}^{x}|\boldsymbol{\eta}^{x})\prod_{i=0}^{n-1}q^{\gamma_{i}^{x}\xi_{[0,n-2-i]}^{x-1}}\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{n-1-i}^{x}}\right)_{\xi_{i}^{x-1}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\xi_{i}^{x-1}}}\\\
=\sum_{\boldsymbol{\gamma}^{x-1}\in\mathfrak{X}}\Phi(\boldsymbol{\gamma}^{x-1}|\boldsymbol{\xi}^{x-1})\prod_{i=0}^{n-1}q^{\gamma_{i}^{x-1}\eta_{[0,n-2-i]}^{x}}\frac{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2+\gamma_{i}^{x-1}}\right)_{\eta_{n-1-i}^{x}}}{\left(q^{-\left(N_{x-1}^{-}(\boldsymbol{\xi}_{i})+N_{x}^{+}(\boldsymbol{\eta}_{n-1-i})\right)+1/2}\right)_{\eta_{n-1-i}^{x}}},$
(105)

which is just Proposition 4.10. ∎

Thus concludes the proof of Theorem 3.4.

###  4.9 Proof of Theorem 3.5

Suppose $\boldsymbol{\xi}$ has $n_{1}$ species 0 particles at site $x_{1}$,
$n_{2}$ species 1 particles at site $x_{2}$. The dual process
$\boldsymbol{\eta}$ has $1$ species 0 particles at site $y_{1}$, $1$ species 1
particles at site $y_{2}$, where $\boldsymbol{\eta}$ evolves with total
asymmetry to the left and $\boldsymbol{\xi}$ evolves with total asymmetry to
the right.

With these special case,

$D^{q-TARZP}(\xi,\eta)=q^{-\sum_{x}\xi_{0}^{x}N_{x+1}^{+}(\eta_{0})+\sum_{x}\eta_{0}^{x}N_{x}^{+}(\xi_{0})}\\\
\prod_{x\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-2\xi_{0}^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\xi_{0})+N_{x+1}^{+}(\eta_{1})\right)+1}\right)\\\
\prod_{x\in\Lambda_{L}}{}_{1}\phi_{0}\left(q^{-2\xi_{1}^{x}};q^{2},q^{-2\left(N_{x-1}^{-}(\xi_{1})+N_{x+1}^{+}(\eta_{0})\right)+1}\right)$
(106)

now becomes

$D^{q-TARZP}(\xi,\eta)=q^{-n_{1}\delta_{y_{1}>x_{1}}+n_{1}\delta_{y_{1}\leq
x_{1}}}{}._{1}\phi_{0}\left(q^{-2n_{1}};q^{2},q^{-2\delta_{y_{2}>x_{1}}+1}\right){}_{1}\phi_{0}\left(q^{-2n_{2}};q^{2},q^{-2\delta_{y_{1}>x_{2}}+1}\right)\\\
=q^{-n_{1}\delta_{y_{1}>x_{1}}+n_{1}\delta_{y_{1}\leq
x_{1}}}\left(q^{-2n_{1}-2\delta_{y_{2}>x_{1}}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-2\delta_{y_{1}>x_{2}}+1};q^{2}\right)_{n_{2}}\\\
=\left\\{\begin{array}[]{cc}q^{-n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}&y_{1}>x_{1},y_{2}>x_{1}\\\
q^{-n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}&y_{1}>x_{1},y_{2}\leq
x_{1}\\\
q^{n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}&x_{2}<y_{1}\leq
x_{1},y_{2}>x_{1}\\\
q^{n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}&x_{2}<y_{1}\leq
x_{1},y_{2}\leq x_{1}\\\
q^{n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}&y_{1}\leq
x_{2},y_{2}>x_{1}\\\
q^{n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}&y_{1}\leq
x_{2},y_{2}\leq x_{1}\\\ \end{array}\right.$ (107)

when $x_{1}>x_{2}$, and if $x_{1}<x_{2}$

$D^{q-TARZP}(\xi,\eta)=\left\\{\begin{array}[]{cc}q^{n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}:=\delta_{1}&y_{1}\leq
x_{1},y_{2}>x_{1}\\\
q^{n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}:=\delta_{2}&y_{1}\leq
x_{1},y_{2}\leq x_{1}\\\
q^{-n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}:=\delta_{3}&x_{1}<y_{1}\leq
x_{2},y_{2}>x_{1}\\\
q^{-n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}+1};q^{2}\right)_{n_{2}}:=\delta_{4}&x_{1}<y_{1}\leq
x_{2},y_{2}\leq x_{1}\\\
q^{-n_{1}}\left(q^{-2n_{1}-1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}:=\delta_{5}&y_{1}>x_{2},y_{2}>x_{1}\\\
q^{-n_{1}}\left(q^{-2n_{1}+1};q^{2}\right)_{n_{1}}\left(q^{-2n_{2}-1};q^{2}\right)_{n_{2}}:=\delta_{6}&y_{1}>x_{2},y_{2}\leq
x_{1}\\\ \end{array}\right.$ (108)