# The inhomogeneous $t$-PushTASEP and Macdonald polynomials at $q=1$

Arvind Ayyer, James Martin, Lauren Williams  Arvind Ayyer, Department of
Mathematics, Indian Institute of Science, Bangalore 560012, India.
<EMAIL_ADDRESS>James Martin, Department of Statistics, University of
Oxford, UK<EMAIL_ADDRESS>Lauren Williams, Department of Mathematics,
Harvard University, Cambridge MA, 02138<EMAIL_ADDRESS>
###### Abstract.

We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites
with site-dependent rates $x_{1},\dots,x_{n}$. Let
$\lambda=(\lambda_{1},\dots,\lambda_{n})$ be a partition whose parts represent
the species of the $n$ particles on the ring. We show that for each
composition $\eta$ obtained by permuting the parts of $\lambda$, the
stationary probability of being in state $\eta$ is proportional to the ASEP
polynomial $F_{\eta}(x_{1},\dots,x_{n};q,t)$ at $q=1$; the normalizing
constant (or partition function) is the Macdonald polynomial
$P_{\lambda}(x_{1},\dots,x_{n};q,t)$ at $q=1$. Our approach involves new
relations between the families of ASEP polynomials and of non-symmetric
Macdonald polynomials at $q=1$. We also use _multiline diagrams_ , showing
that a single jump of the PushTASEP system is closely related to the operation
of moving from one line to the next in a multiline diagram. We derive symmetry
properties for the system under permutation of its jump rates, as well as a
formula for the current of a single-species system.

###### Key words and phrases:

Macdonald polynomials, PushTASEP, ASEP polynomials, exclusion process,
multiline diagram, permuted basement Macdonald polynomials, nonsymmetric
Macdonald polynomials

###### 2010 Mathematics Subject Classification:

05A10, 05A19, 05A05, 05E05, 05E10, 33D52, 60J10, 60K35

###### Contents

  1. 1 Introduction
  2. 2 Basic properties of the $t$-PushTASEP
  3. 3 Background on Macdonald and ASEP polynomials
  4. 4 Nonsymmetric Macdonald and ASEP polynomials at $q=1$
  5. 5 Multiline diagrams
  6. 6 Proof of 1.1
  7. 7 Formulas for density and currents

##  1\. Introduction

Multispecies versions of the asymmetric simple exclusion process (ASEP) and
its relatives have been the subject of intense study in recent years, from
diverse perspectives in physics, probability, algebra, and combinatorics. The
connection between the multispecies ASEP on a ring and Macdonald polynomials
was developed by Cantini, de Gier and Wheeler [CdGW15] and Chen, de Gier and
Wheeler [CdGW20]. In these works, they define the family of ASEP polynomials,
which are polynomials in variables $x_{1},\dots,x_{n}$ whose coefficients are
rational functions in $q$ and $t$. When specialised to $q=1$ and
$x_{1}=x_{2}=\dots=x_{n}$, the ASEP polynomials describe the stationary
distribution of a multispecies ASEP on a ring with $n$ sites. The ASEP
polynomials are in fact special cases of the _permuted-basement Macdonald
polynomials_ introduced in [Fer11], as shown in [CMW22].

A construction of the stationary distribution of the multispecies ASEP in
terms of multiline diagrams was given in [Mar20], building on the construction
for the TASEP by Ferrari and Martin [FM07] and the matrix product
representation for the ASEP given by Prolhac, Evans and Mallick [PEM09].
Corteel, Mandelshtam and Williams [CMW22] then showed that a generalisation of
the multiline diagrams from [Mar20] could be used to give a combinatorial
formula for the ASEP polynomials with general $x_{1},\dots,x_{n}$, $q$ and
$t$.

The description of the mASEP stationary distribution in terms of ASEP
polynomials with identical $x_{i}$ invites the question: is there a natural
multitype particle system, with inhomogeneous (i.e. site-dependent) jump
rates, whose stationary probabilities are given by the ASEP polynomials with
general $x_{i}$? An inhomogeneous version of the ASEP itself is not believed
to have nice algebraic properties. The main result of this article is that a
related process, the _multispecies $t$-PushTASEP with inhomogeneous rates_,
does indeed have its stationary distribution given by the ASEP polynomials
with general $x_{i}$ – see 1.1.

Our approach involves new relations between the families of ASEP polynomials
and of non-symmetric Macdonald polynomials at $q=1$, building on the work of
Alexandersson and Sawhney [AS19]. Among other results, we show that certain
ratios of non-symmetric Macdonald polynomials become symmetric in the
particular case $q=1$. We also use the multiline diagram construction – a
single jump of the PushTASEP system is closely related to the operation moving
from one line to the next in a multiline diagram.

Systems related to the multispecies $t$-PushTASEP have previously appeared in
various contexts. The multispecies system in the case $t=0$ with homogenous
rates was already studied by Ferrari and Martin [FM06], under the name of
_discrete-space Hammersley-Aldous-Diaconis process_ (or long-range exclusion
process [Spi70]). A related process in discrete time (dubbed the “frog model”)
defined on the ring was recently used by Bukh and Cox to study problems
involving the longest common subsequence between a periodic word and a word
with i.i.d. uniform entries. Moving to the inhomogeneous case, a single-type
PushTASEP in the case $t=0$ on the half-line was considered by Petrov [Pet20],
and the multi-type $t>0$ case on a finite interval has been investigated by
Borodin and Wheeler [BW22, Section 12.5] in the context of the coloured
stochastic six-vertex model

Most recently, in independent work, Aggarwal, Nicoletti and Petrov [ANP23]
obtain closely related results. They write the stationary distribution of the
multitype inhomogeneous PushTASEP (and other related models including the
mASEP and the multi-type TAZRP) in terms of vertex models, which are closely
related to multiline diagrams and to matrix product formulae. Their approach
is entirely different to ours, making extensive use of Yang-Baxter interchange
relations.

In a companion paper [AM23], two of the authors focus on the particular case
$t=0$. We employ more direct probabilistic methods involving time-reversal and
coupling to connect the stationary distribution to multiline diagrams, and we
describe symmetry properties under permutation of the rates, which apply to
evolutions of the system out of equilibrium as well as to the stationary
distribution.

###  1.1. Definition of the $t$-PushTASEP

In this paper we study the inhomogeneous $t$-PushTASEP on a ring with $n$
sites, which generalizes the PushTASEP studied in [AM23].

A configuration of the system is a vector (or _composition_)
$(\eta_{1},\dots,\eta_{n})$ whose entries are non-negative integers. The entry
$\eta_{j}$ denotes the species of the particle at site $j$. If two particles
have species $i_{1}$ and $i_{2}$ with $i_{1}>i_{2}$, we say that the particle
of species $i_{1}$ is _stronger_ and the particle of species $i_{2}$ is
_weaker_. We often refer to particles of species $0$ as _holes_ or
_vacancies_.

The $t$-PushTASEP dynamics will preserve the number of particles of each
species, so we may take the state-space of the system to be
$S_{\lambda}=S_{n}(\lambda)$, the set of compositions which are permutations
of some given _partition_ $\lambda=(\lambda_{1},\dots,\lambda_{n})$ with
$\lambda_{1}\geq\dots\geq\lambda_{n}\geq 0$. We can describe such a partition
by its vector of types $\mathbf{m}=(m_{0},m_{1},\dots,m_{s})$, where
$m_{i}=\\#\\{j:\lambda_{j}=i\\}$ gives the number of particles of species $i$,
and where $s$ is the largest species in the system. Sometimes we denote our
partition by $\lambda=\langle s^{m_{s}},\dots,1^{m_{1}},0^{m_{0}}\rangle$. We
have $\sum_{i=0}^{s}m_{i}=n$. We will always require that $m_{0}\geq 1$, i.e.
that the system has at least one vacancy. The partition $\lambda$ is called
the _content_ of the system.

The system has positive real parameters $x_{1},\dots,x_{n}$. We first define
the transitions of the PushTASEP, i.e. the case $t=0$. For each site $j$,
there is an exponential clock which rings with rate $1/x_{j}$. The effect of a
bell ringing at site $j$ is as follows. If site $j$ contains a vacancy then
nothing changes. If instead site $j$ contains a particle of species $r_{0}>0$,
this particle becomes “active”. It moves clockwise around the ring until it
finds a site $j_{1}$ with a particle of smaller species $r_{1}<r_{0}$. The
active particle now settles at site $j_{1}$. If in fact $r_{1}=0$ (i.e. the
site $j_{1}$ was previously vacant) then the procedure stops; otherwise the
particle of label $r_{1}$ becomes active and itself starts to move clockwise
around the ring looking for a site with a particle of smaller species
$r_{2}<r_{1}$. Such a procedure continues until a vacancy is found. All the
transitions occur simultaneously and the original site $j$ becomes vacant at
the end of the transition.

The $t$-PushTASEP is a generalization of the PushTASEP, with an additional
parameter $t$ which for convenience we take to be in $[0,1)$ (though it is
easy to extend to $t\geq 1$). Again each site $j$ has a bell ringing at rate
$1/x_{j}$, and we describe the effect of such a bell. If site $j$ is vacant
then nothing changes. Otherwise, as above, the particle of type $r_{0}>0$ at
site $j$ becomes “active” and will move to the location of a weaker particle.
However, for $t>0$ the move is not deterministic. Suppose there are $m$
particles in the system whose species is less than $r_{0}$ (including
vacancies). Recall that $[m]_{t}=1+t+\cdots+t^{m-1}=\frac{1-t^{m}}{1-t}$
denotes the $t$-analogue of the integer $m$. Then the particle at site $j$
will travel clockwise around the ring, and with probability $t^{k-1}/[m]_{t}$,
it will move to the location of the $k$’th of these lower-species particles.
If this location is not vacant, then the particle there becomes active, and
chooses a weaker particle to displace in the same way. The procedure continues
until a vacancy is chosen. All these transitions occur simultaneously. Again
the site $j$ itself always becomes vacant at the end of the transition.

$\begin{array}[]{|c|c|}\hline\cr\text{Configuration}&\text{Rate}\\\
\hline\cr&\\\\[-5.69046pt]
(2,4,\underline{0},\underline{3},2,4,1,3)&\displaystyle\frac{1}{x_{3}}\frac{1}{[4]_{t}}\\\\[14.22636pt]
(2,4,\underline{0},\underline{1},\underline{3},4,\underline{2},3)&\displaystyle\frac{1}{x_{3}}\frac{t}{[4]_{t}}\frac{1}{[2]_{t}}\\\\[14.22636pt]
(2,4,\underline{0},\underline{2},\underline{3},4,1,3)&\displaystyle\frac{1}{x_{3}}\frac{t}{[4]_{t}}\frac{t}{[2]_{t}}\\\\[14.22636pt]
(2,4,\underline{0},\underline{1},2,4,\underline{3},3)&\displaystyle\frac{1}{x_{3}}\frac{t^{2}}{[4]_{t}}\\\\[14.22636pt]
(\underline{3},4,\underline{0},\underline{2},2,4,1,3)&\displaystyle\frac{1}{x_{3}}\frac{t^{3}}{[4]_{t}}\frac{1}{[2]_{t}}\\\\[14.22636pt]
(\underline{3},4,\underline{0},\underline{1},2,4,\underline{2},3)&\displaystyle\frac{1}{x_{3}}\frac{t^{3}}{[4]_{t}}\frac{t}{[2]_{t}}\\\\[14.22636pt]
\hline\cr\end{array}$

Figure 1.  Let $\eta=(2,4,3,0,2,4,1,\allowbreak 3)$ with $n=8$ and $s=4$. If
the bell rings at site $3$, some particles will move – the table shows the
possible destination configurations, along with the rate of the jump to each
one. In each case the particles which moved are underlined. The transition
corresponding to the $4$th line of the table is illustrated on the right. Site
$1$ is shown at the top of the ring, and site $3$ where the bell rings is on
the extreme right.

We may interpret the procedure above by saying that the active particle moves
clockwise around the ring looking for a weaker particle to displace, but
rejects each option with probability $t$. See Section 4.1 for an equivalent
definition along these lines.

See Figure 1 for examples of transitions from a given configuration on a ring
of size $8$. The state diagram for the system defined by $\lambda=(2,1,0)$,
i.e. $\mathbf{m}=(1,1,1)$, is given in Figure 2.

Figure 2. The transition graph of the multispecies $t$-PushTASEP for
$\mathbf{m}=(1,1,1)$.

###  1.2. Main results

Our main result is a description of the stationary distribution of a
$t$-PushTASEP system in terms of the _ASEP polynomials_ , cf. 3.9.

######  Theorem 1.1.

In the multispecies $t$-PushTASEP with content
$\lambda=(\lambda_{1},\dots,\lambda_{n})$ and parameters
$\mathbf{x}=(x_{1},\dots,x_{n})$, the stationary probability of a
configuration $\eta\in S_{\lambda}$ is given by

$\pi_{\lambda}(\eta)=\frac{{F}_{\eta}(\mathbf{x};1,t)}{P_{\lambda}(\mathbf{x};1,t)},$

where ${F}_{\eta}(\mathbf{x};q,t)$ is the ASEP polynomial from 3.9 associated
to $\eta$, and $P_{\lambda}(\mathbf{x};q,t)$ is the Macdonald polynomial
associated to $\lambda$.

######  Example 1.2.

The steady state probabilities for the example from Figure 2 are proportional
to the ASEP polynomials $F_{\eta}$ at $q=1$, which are given in the following
table. (3.12 shows the same ASEP polynomials but for general $q$.) The sum of
these polynomials is the Macdonald polynomial

$P_{(2,1,0)}(x_{1},x_{2},x_{2};1,t)=\left(x_{1}+x_{2}+x_{3}\right)\left(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}\right),$

which can be seen as a partition function for the system; note that it is
independent of $t$ (see (3.10)).

$\begin{array}[]{|c|c|}\hline\cr\eta&{F}_{\eta}(x_{1},x_{2},x_{2};1,t)\\\
\hline\cr(2,1,0)&\displaystyle
x_{1}x_{2}\left(x_{1}+\frac{x_{3}}{1+t}\right)\\\ (2,0,1)&\displaystyle
x_{1}x_{3}\left(x_{1}+\frac{x_{2}t}{1+t}\right)\\\ (1,2,0)&\displaystyle
x_{1}x_{2}\left(x_{2}+\frac{x_{3}t}{1+t}\right)\\\ (1,0,2)&\displaystyle
x_{1}x_{3}\left(\frac{x_{2}}{1+t}+x_{3}\right)\\\ (0,2,1)&\displaystyle
x_{2}x_{3}\left(\frac{x_{1}}{1+t}+x_{2}\right)\\\ (0,1,2)&\displaystyle
x_{2}x_{3}\left(\frac{x_{1}t}{1+t}+x_{3}\right)\\\ \hline\cr\end{array}$

A combinatorial formula for the ASEP polynomials in terms of multiline
diagrams was given in [CMW22]. Combining this with 1.1, we obtain the
following corollary.

######  Corollary 1.3.

Consider a multiline diagram as defined in [CMW22] of type $\lambda$, with
parameters $x_{1},\dots,x_{n},t$ and $q=1$. The distribution of the bottom
line of the diagram is the same as the stationary distribution of the
$t$-PushTASEP with content $\lambda$.

We will define multiline diagrams in Section 5.2 for the special case where
$\lambda$ has all parts distinct. See [CMW22] for the general definition.

From 1.1, we can also derive a symmetry property for the $t$-PushTASEP under
permutation of the jump-rate parameters $x_{i}$. If $O$ is an observable (an
event or a random variable), write $\langle O\rangle$ for its probability or
expectation in the stationary distribution.

######  Theorem 1.4.

Fix $k<n$ and let $O$ be any observable in the stationary distribution of
multispecies $t$-PushTASEP which depends only on the configuration in sites
$1,2,\dots,k$. Then $\langle O\rangle$ is symmetric in the parameters
$x_{k+1},\dots,x_{n}$.

Note that in [AM23], a symmetry result which is stronger than 1.4 is proved in
the case $t=0$. That result extends also to observables depending on the path
of the process (not just its state at a single time), and also to processes
out of equilibrium (if started from suitable initial states). Whether this
stronger symmetry property also holds for $t>0$ is an interesting open
question (see e.g. [AMM22] for related discussions in the case of the totally
asymmetric zero-range process).

We now consider other important quantities for the $t$-PushTASEP in its
stationary distribution. Two natural such quantities are the density of a
particular species, which is the probability of seeing a particle of that
species at some site, and the current, which is the number of particles of a
given species crossing an edge per unit time.

We will show in 7.1 that the formulas for the density are independent of $t$,
and hence the same as that for $t=0$ given in [AM23]. The case of the current
is much more interesting. It turns out that even when we have only a single
species, the formula is nontrivial. We will prove the following result.

######  Theorem 1.5.

The current between two adjacent sites (say sites $n$ and $1$) in the
stationary distribution of the single species $t$-PushTASEP on $S_{\langle
1^{m_{1}},0^{m_{0}}\rangle}$ is given by

$\displaystyle J_{m_{0},m_{1}}$
$\displaystyle=\frac{1+2t+3t^{2}+\cdots+m_{0}t^{m_{0}-1}}{1+t+\cdots+t^{m_{0}-1}}\cdot\frac{e_{m_{1}-1}(x_{1},\dots,x_{n})}{e_{m_{1}}(x_{1},\dots,x_{n})}$
$\displaystyle=\frac{\frac{d}{dt}(1+t+t^{2}+\cdots+t^{m_{0}})}{1+t+\cdots+t^{m_{0}-1}}\cdot\frac{e_{m_{1}-1}(x_{1},\dots,x_{n})}{e_{m_{1}}(x_{1},\dots,x_{n})},$

The $t=0$ case of this result was proved in [AM23] using a coloring argument.
We will prove the result in Section 7. Generalizing the formula for the
current to the multispecies $t$-PushTASEP seems considerably harder, and in
Section 7 we explain why the coloring approach does not work in that case.

The structure of this paper is as follows. In Section 2 we discuss some basic
properties of the $t$-PushTASEP, including an important recoloring property.
In Section 3 we provide background on nonsymmetric Macdonald polynomials, ASEP
polynomials, (symmetric) Macdonald polynomials, and permuted basement
Macdonald polynomials. In Section 4 we prove some properties of nonsymmetric
Macdonald polynomials and ASEP polynomials at the specialization $q=1$, in
particular 4.18, which will be a main ingredient in our proof of 1.1. In
Section 5 we define multiline diagrams and explain their relation to the
$t$-PushTASEP. In Section 6 we prove our main result, 1.1. We end the paper
with the formulas for the density and the current in stationarity in Section
7.

### Acknowledgements

We thank Per Alexandersson, Gidi Amir, Luigi Cantini, Pablo Ferrari, Jan de
Gier, Svante Linusson, Leo Petrov, and Michael Wheeler for helpful
discussions. We especially thank Omer Angel for valuable discussions while
this project was in its formative stages. AA acknowledges support from the DST
FIST program - 2021 [TPN - 700661]. and by SERB Core grant CRG/2021/001592. LW
is supported by the National Science Foundation under Award No. DMS-2152991.
Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the National Science Foundation.

##  2\. Basic properties of the $t$-PushTASEP

In this section we discuss some basic properties of the $t$-PushTASEP system
which we will rely on in the later analysis.

###  2.1. Projections and couplings

The multispecies dynamics defined above have an important “recolouring”
property. If we relabel the particles while (weakly) preserving the order of
the labels, the resulting system still follows $t$-PushTASEP dynamics. This
allows us to project from a “finer” multispecies system to a “coarser” one, by
merging groups of two or more adjacent species into one.

As an extreme case, we can consider all particles of species $i,\dots,s$ as
“particles” (with the new label $1$) and all particles of species
$1,\dots,i-1$ as vacancies (with the new label $0$), to obtain a single-
species process with a total of $a_{i}:=m_{i}+\dots+m_{s}$ particles and
$n-a_{i}$ vacancies. Considering such projections for all $i=1,2,\dots,s$, we
can identify the multispecies process as a coupling of $s$ single-species
processes. This is a version of the basic coupling [Lig85, Chapter VIII,
Section 2] (under which the bells ring at the same sites at the same types in
all the coupled single-species systems).

To state the above projection (or lumping) properties precisely, we make the
following definition.

######  Definition 2.1.

We say that a function $\phi$ from $\mathbb{N}$ to $\mathbb{N}$ is _weakly
order-preserving_ if $\phi(i)\leq\phi(j)$ whenever $i\leq j$. For such a
function $\phi$ and a composition $\rho=(\rho_{1},\dots,\rho_{n})$, define
$\phi(\rho)$ componentwise by
$\phi(\rho)=(\phi(\rho_{1}),\dots,\phi(\rho_{n}))$.

For example, the function $\phi$ which sends elements of $\\{0,1,2,3,4\\}$ to
$2$, elements of $\\{5,6\\}$ to $4$, and is the identity otherwise, is a
weakly order-preserving function. Note that if $\rho$ is a partition then so
is $\phi(\rho)$.

######  Proposition 2.2.

Let $\phi:\mathbb{N}\mapsto\mathbb{N}$ be a weakly order-preserving function
with $\phi(0)=0$. Consider a multispecies $t$-PushTASEP process with content
given by the partition $\lambda$. Via the map $\phi$, this process projects to
a multispecies $t$-PushTASEP with content $\mu$, where $\mu=\phi(\lambda)$.

###### Proof.

Recall the description of the $t$-PushTASEP dynamics from Section 1.1.
Consider an active particle of species $r$, in a system with $m$ particles
weaker than $r$ (including vacancies). For $1\leq k\leq m$, the particle moves
to the $k$th out of the $m$ locations containing such a weaker particle
(considered in order clockwise from its current location) with probability
$\frac{t^{k-1}}{1+t+\dots+t^{m-1}}$ – it displaces the particle currently
occupying that site, which itself becomes active.

We may alternatively describe the procedure as follows. The particle moves
clockwise around the ring, and each time it passes a site with a weaker
particle, it settles at that site with probability $1-t$, and continues moving
with probability $t$. If it passes the $m$th such site, then it continues
cyclically around the ring, with the $(m+1)$st option it considers being the
same as the first, and so on. Hence for $1\leq k\leq m$, it chooses the $k$th
available option with probability

$\displaystyle(1-t)(t^{k-1}+t^{k-1+m}+t^{k-1+2m}+\dots)$
$\displaystyle=(1-t)t^{k-1}(1+t^{m}+t^{2m}+\dots)$
$\displaystyle=\frac{t^{k-1}}{1+t+\dots+t^{m-1}},$

as in our original description of the procedure.

We make use of one further freedom – when the active particle passes a site
containing a particle with the same label, it makes no difference whether we
allow the active particle to displace its “twin” or not.

This description allows us to maintain a coupling between a system of
particles $(\eta(u),u\geq 0)$ with content $\lambda$ and a system of particles
$(\zeta(u),u\geq 0)$ with content $\mu$, such that at all times $u\geq 0$,
$\zeta(u)=\phi(\eta(u))$, i.e. $\zeta_{j}(u)=\phi(\eta_{j}(u))$ for all $1\leq
j\leq n$. The bells ring at the same time in both systems. When a bell rings
at some site in the $\lambda$-system currently in configuration $\eta$, we
observe some collection of transitions of particles according to the
description above, with each active particle moving clockwise and settling on
an available location with probability $1-t$. Under the coupling, if a
particle from some site $j$ moves to some new site $j^{\prime}$ in the
$\lambda$-system, exactly the same will occur in the $\mu$-system. Since
$\eta(j)>\eta(j^{\prime})$ and $\phi$ is weakly order-preserving, we have that
$\phi(\eta(j))\geq\phi(\eta(j^{\prime}))$, and the same transition is possible
in the $\mu$-system as required.

Note that this can also be interpreted as a commutation property. Let $\eta\in
S_{\lambda}$ be a configuration of the $\lambda$-system, and let $j$ be any
site. Consider the follow two operations to obtain $\zeta\in S_{\mu}$: (a)
generate a configuration $\eta^{\prime}$ resulting from the ring of a bell at
site $j$, and then recolour $\eta^{\prime}$ by $\phi$ to obtain a state
$\zeta=\phi(\eta^{\prime})$; (b) recolour $\eta$ to give
$\zeta^{\prime}=\phi(\eta)$, and then generate a configuration $\zeta$ from
$\zeta^{\prime}$ resulting from the ring of a bell at site $j$. The coupling
above shows that (a) and (b) lead to the same distribution of $\zeta\in
S_{\mu}$. This commutation gives the required projection property. ∎

We can then immediately deduce a corresponding recolouring property for the
stationary distributions:

######  Proposition 2.3.

Let $\phi:\mathbb{N}\mapsto\mathbb{N}$ be a weakly order-preserving function
with $\phi(0)=0$, and suppose we have partitions $\mu$ and $\lambda$ with
$\phi(\lambda)=\mu$. Let $\pi_{(\lambda)}=(\pi_{(\lambda)}(\eta)$, $\eta\in
S_{\lambda})$ and $\pi_{(\mu)}=(\pi_{(\mu)}(\eta)$, $\eta\in S_{\mu})$ denote
the stationary distributions of the $t$-PushTASEP with content $\lambda$ and
$\mu$. Then for all $\eta\in S_{\mu}$,

(2.1) $\pi_{(\mu)}(\eta)=\sum_{\zeta\in
S_{\lambda}:\phi(\zeta)=\eta}\pi_{(\lambda)}(\zeta).$

###  2.2. Transition rates

For completeness and later use, we give here a direct description of the
transition rates for the $t$-PushTASEP dynamics. Let $\lambda$ be a partition,
and as above let $m_{i}$ be the number of entries $i$ in $\lambda$.

Let $\eta,\eta^{\prime}\in S_{\lambda}$. It follows from the description in
Section 1.1 that a bell at $j$ can cause a transition from the configuration
$\eta$ to the configuration $\eta^{\prime}$ precisely if the following
conditions are satisfied:

  * •

$j$ is the unique site which is vacant in $\eta^{\prime}$ and not vacant in
$\eta$. For every other site $i$, $\eta^{\prime}(i)\geq\eta(i)$.

  * •

For each type $h>0$ with $m_{h}>0$, either:

    1. (1)

the sites occupied by species $h$ are the same in $\eta$ and $\eta^{\prime}$;
or,

    2. (2)

There exists exactly one site $j(h)$ such that $\eta_{j(h)}=h$ and
$\eta^{\prime}_{j(h)}\neq h$. It follows that there also exists exactly one
site $j^{\prime}(h)$ such that $\eta^{\prime}_{j^{\prime}(h)}=h$ and
$\eta_{j^{\prime}(h)}\neq h$.

Define $w_{\eta,\eta^{\prime}}(h)$ for each $h$ as follows. If case (1) holds
then $w_{\eta,\eta^{\prime}}(h)=1$. If case (2) holds then let $K_{h}$ be the
number of entries of $\lambda$ smaller than $h$ (including zeros). Let
$\ell_{h}$ be the number of sites in the cyclic interval
$(j(h),j^{\prime}(h))$, excluding endpoints, with value smaller than $h$ in
$\eta^{\prime}$. Let

$w_{\eta,\eta^{\prime}}(h)=\frac{t^{\ell_{h}}}{1+t+\dots+t^{K_{h}-1}}.$

Suppose the system is in state $\eta$. When a bell rings at $j$, a jump occurs
to $\eta^{\prime}$ with probability

(2.2) $\prod_{h>0:m_{h}>0}w_{\eta,\eta^{\prime}}(h).$

(The species $h$ for which case (2) holds above are precisely those for which
some particle of species $h$ becomes “active” during the transition, in the
sense of Section 1.1.) The transition rate from $\eta$ to $\eta^{\prime}$ is
therefore

$\frac{1}{x_{j}}\prod_{h>0:m_{h}>0}w_{\eta,\eta^{\prime}}(h).$

###  2.3. Single-species stationary distributions

In the case that we have only one species of particle, it turns out that the
stationary distribution is independent of $t$. It thus matches the
distribution when $t=0$ given in [AM23].

######  Proposition 2.4.

Let $\lambda=\langle 1^{m_{1}},0^{m_{0}}\rangle$ where $m_{1}+m_{0}=n$, and
define

(2.3)
$\pi(\eta):=\frac{1}{e_{m_{1}}(x_{1},\dots,x_{n})}\prod_{\begin{subarray}{c}i=1\\\
\eta_{i}=1\end{subarray}}^{n}x_{i}.$

Then the stationary probability of $\eta\in S_{\lambda}$ for the $t$-PushTASEP
is $\pi(\eta)$.

###### Proof.

Since the $t$-PushTASEP is irreducible, it suffices to verify the global
balance equations. The total weight of outgoing transitions from the
configuration $\eta$ which involve a particle at site $j$ is
$\pi(\eta)/x_{j}$, since the particle at site $j$ makes a transition to some
vacancy with rate $1/x_{j}$.

As for the incoming rate to $\eta$, note that a configuration $\tau$ makes a
transition to $\eta$ if there exist a pair of positions $j\neq k$ such that
$\eta_{j}=1$, $\eta_{k}=0$, $\tau_{j}=0$, and $\tau_{k}=1$, and $\eta$ and
$\tau$ agree outside of positions $j$ and $k$. Then
$\pi(\eta)/\pi(\tau)=x_{j}/x_{k}$. If there are $a$ vacancies strictly between
positions $k$ and $j$ (traveling in the clockwise direction starting at $k$),
then the weight of the transition from $\tau$ to $\eta$ is
$\pi(\tau)\cdot\frac{1}{x_{k}}\cdot\frac{t^{a}}{[m_{0}]_{t}}=\pi(\eta)\cdot\frac{1}{x_{j}}\cdot\frac{t^{a}}{[m_{0}]_{t}}.$
But now if we fix $j$ and sum over all possible $k$ (and corresponding
$\tau$), the weight of all these transitions to $\eta$ will be
$\pi(\eta)\cdot\frac{1}{x_{j}}$. This is exactly the same as the total weight
of outgoing transitions from $\eta$ involving the particle at site $j$, as
argued above. Summing over all possible locations of particles completes the
proof. ∎

2.4 will be useful when we discuss the generation of multiline diagrams in
Section 5.2. We will also use this result to analyze the density of particles
in Section 7.

##  3\. Background on Macdonald and ASEP polynomials

In this section we define nonsymmetric Macdonald polynomials, ASEP
polynomials, and (symmetric) Macdonald polynomials. We also mention some
relations with permuted basement Macdonald polynomials. All of the above
polynomials are elements of the polynomial ring
$\mathbb{Q}(q,t)[x_{1},\dots,\allowbreak x_{n}]$ in variables
$x_{1},\dots,x_{n}$, with coefficients in $\mathbb{Q}(q,t)$.

###  3.1. Nonsymmetric Macdonald polynomials

Nonsymmetric Macdonald polynomial can be defined as eigenfunctions of the _$q$
-Dunkl_ or _Cheredik operators_. We will mostly follow the notation of
[CMW22].

For $f:=f(x_{1},\dots,x_{n};q,t)\in\mathbb{R}$, we define the operators
$s_{i}$ and $L_{i}$, $1\leq i\leq n-1$ as

$s_{i}(f)=f(x_{1},\dots,x_{i-1},x_{i+1},x_{i},x_{i+2},\dots,x_{n})$

and

$L_{i}(f)=\frac{tx_{i}-x_{i+1}}{x_{i}-x_{i+1}}\left(f-s_{i}(f)\right).$

Using these, we define operators $T_{i}$ and $T_{i}^{-1}$, $1\leq i\leq n-1$,
as

(3.1) $T_{i}(f)=tf-L_{i}(f),\qquad T_{i}^{-1}(f)=t^{-1}f-t^{-1}L_{i}(f).$

These operators satisfy the _Hecke algebra_ relations,

(3.2) $\begin{split}(T_{i}-t)(T_{i}+1)=0,&\quad
T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},\\\ T_{i}T_{j}=T_{j}T_{i}&\quad\text{if
$|i-j|\geq 1$.}\end{split}$

We also define the shift operator $\omega$ as

(3.3) $(\omega f)(x_{1},\dots,x_{n})=f(qx_{n},x_{1},\dots,x_{n-1}).$

Using the operators $T_{i},T_{i}^{-1}$ and $\omega$ we define the _Cherednik-
Dunkl operators_ $Y_{i}$, $1\leq i\leq n$ by

$Y_{i}=t^{-(i-1)}T_{i}^{-1}\cdots T_{n-1}^{-1}\omega T_{1}\cdots T_{i-1}.$

One can show that these $Y_{i}$’s mutually commute and can therefore be
simultaneously diagonalized.

######  Remark 3.1.

Let $h,f\in\mathbb{Q}(q,t)[x_{1},\dots,\allowbreak x_{n}]$ such that $h$ is
symmetric in $x_{1},\dots,x_{n}$. It follows from the definition of the
operator $T_{i}$ that $T_{i}(hf)=hT_{i}(f)$ for all $i$.

A (weak) _composition_ $\eta=(\eta_{1},\dots,\eta_{n})$ is a tuple of
nonnegative integers. Let

(3.4) $y_{i}(\eta;q,t)=q^{\eta_{i}}\,t^{-|\\{j>i\ |\
\eta_{j}\geq\eta_{i}\\}|-|\\{j<i\ |\ \eta_{j}>\eta_{i}\\}|}.$

######  Definition 3.2.

Let $\mathcal{P}_{n}$ denote the set of partitions of $n$. The _dominance
order_ on partitions, denoted $\leq$, is a partial order on the partitions in
$\mathcal{P}_{n}$ with fixed size, defined as follows. We say that
$\lambda\leq\mu$ if $\lambda_{1}+\cdots+\lambda_{i}\leq\mu_{1}+\cdots+\mu_{i}$
for $1\leq i\leq n$.

To extend the dominance order to an order on compositions $\eta$, let
$\eta^{+}$ be the partition obtained by ordering the parts of $\eta$ in weakly
decreasing order. We write $\eta\leq\nu$ for compositions $\eta$ and $\nu$ if
either $\eta^{+}\leq\nu^{+}$, or $\eta^{+}=\nu^{+}$ and
$\eta_{1}+\cdots+\eta_{i}\leq\nu_{1}+\cdots+\nu_{i}$ for $1\leq i\leq n$. Note
that this partial order is not the natural generalization of the dominance
order to compositions.

######  Definition 3.3.

[Che95, Mar99] The _nonsymmetric Macdonald polynomial_
$E_{\eta}=E_{\eta}(\mathbf{x};q,t)$ associated to a composition
$\eta=(\eta_{1},\dots,\eta_{n})$ is the polynomial uniquely defined by the
conditions

  1. (1)

$E_{\eta}(\mathbf{x};q,t)$ has the monomial expansion

$E_{\eta}(\mathbf{x};q,t)=\sum_{\zeta\leq\eta}v_{\eta,\zeta}(q,t)\mathbf{x}^{\zeta},$

where $\mathbf{x}^{\zeta}$ is shorthand for $x_{1}^{\zeta_{1}}\cdots
x_{n}^{\zeta_{n}}$, and $v_{\zeta,\zeta}=1$.

  2. (2)

$Y_{i}E_{\eta}=y_{i}(\eta;q,t)E_{\eta}$ for all $1\leq i\leq n$ and all
compositions $\eta$.

The existence of these polynomials is highly nontrivial.

######  Remark 3.4.

It follows from (3.4) that two distinct nonsymmetric Macdonald polynomials
$E_{\eta}$ and $E_{\tau}$ have distinct tuples of eigenvalues
$(y_{1},\dots,y_{n})$ as functions of $q$ and $t$. However, they may not be
distinct as functions of $t$ when we set $q=1$; then for example, $E_{210}$,
$E_{211}$, and $E_{100}$ all have the same eigenvalues.

However, if $\eta$ and $\tau$ are permutations of the same partition then it
still does hold that the tuples of their eigenvalues remain distinct under the
specialisation $q=1$.

######  Example 3.5.

Consider all permutations of the partition $(2,1,0)$ as compositions. The
nonsymmetric Macdonald polynomials for each of them, along with their
eigenvalues, are as follows:

$\eta$ | $E_{\eta}(x_{1},x_{2},x_{3})$ | $\scriptstyle y_{1}(\eta)$ | $\scriptstyle y_{2}(\eta)$ | $\scriptstyle y_{3}(\eta)$
---|---|---|---|---
$(0,1,2)$ |  $\frac{(t-1)\left(q^{2}t^{3}-2qt^{2}+t^{2}-t+1\right)}{(qt-1)^{3}(qt+1)}x_{1}^{2}x_{2}+\frac{(t-1)^{2}}{(qt-1)^{2}}(x_{1}x_{2}^{2}+x_{1}^{2}x_{3})$ $+x_{2}x_{3}^{2}+\frac{(t-1)\left(q^{3}t^{3}+2q^{2}t^{3}-3q^{2}t^{2}-2qt+q-t+2\right)}{(qt-1)^{3}(qt+1)}x_{1}x_{2}x_{3}$ $+\frac{1-t}{1-qt}(x_{2}^{2}x_{3}+x_{1}x_{3}^{2})$ | $t^{-2}$ | $qt^{-1}$ | $q^{2}$
$(0,2,1)$ |  $\frac{(t-1)^{2}}{(qt-1)^{2}(qt+1)}x_{1}^{2}x_{2}+\frac{1-t}{1-qt}x_{1}x_{2}^{2}+\frac{1-t}{1-q^{2}t^{2}}x_{1}^{2}x_{3}$ $+\frac{(t-1)\left(q^{2}t^{2}+qt-q-1\right)}{(qt-1)^{2}(qt+1)}x_{1}x_{2}x_{3}+x_{2}^{2}x_{3}$ | $t^{-2}$ | $q^{2}$ | $qt^{-1}$
$(1,0,2)$ |  $\frac{(t-1)^{2}}{(qt-1)^{2}(qt+1)}x_{1}^{2}x_{2}+\frac{1-t}{1-q^{2}t^{2}}x_{1}x_{2}^{2}+\frac{1-t}{1-qt}x_{1}^{2}x_{3}$ $+\frac{(t-1)\left(q^{2}t^{2}+qt-q-1\right)}{(qt-1)^{2}(qt+1)}x_{1}x_{2}x_{3}+x_{1}x_{3}^{2}$ | $qt^{-1}$ | $t^{-2}$ | $q^{2}$
$(1,2,0)$ | $\frac{1-t}{1-qt}x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+\frac{q(1-t)}{1-qt}x_{1}x_{2}x_{3}$ | $qt^{-1}$ | $q^{2}$ | $t^{-2}$
$(2,0,1)$ | $\frac{1-t}{1-qt}x_{1}^{2}x_{2}+x_{1}^{2}x_{3}+\frac{q(1-t)}{1-qt}x_{1}x_{2}x_{3}$ | $q^{2}$ | $t^{-2}$ | $qt^{-1}$
$(2,1,0)$ | $x_{1}^{2}x_{2}+\frac{q(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}$ | $q^{2}$ | $qt^{-1}$ | $t^{-2}$

###  3.2. ASEP polynomials

Cantini, de Gier and Wheeler [CdGW15] related Macdonald polynomials
$P_{\lambda}$ at $x_{i}=1$ and $q=1$ to the multispecies ASEP on a ring, via
the notion of a _qKZ family_ , which we now explain.

The following notion of _qKZ family_ was introduced in [KT07], also explaining
the relationship of such polynomials to nonsymmetric Macdonald polynomials. We
use the conventions of [CdGW15, Section 1.3].

######  Definition 3.6.

Fix a partition $\lambda=(\lambda_{1},\dots,\lambda_{n})$. We say that a
family $\\{f_{\eta}\\}_{\eta\in S_{\lambda}}$ of homogeneous degree
$|\lambda|$ polynomials in $n$ variables $\mathbf{x}=(x_{1},\dots,x_{n})$,
with coefficients which are rational functions of $q$ and $t$, is a _qKZ
family_ if they satisfy

(3.5) $\displaystyle T_{i}f_{\eta}(\mathbf{x};q,t)$
$\displaystyle=f_{s_{i}\eta}(\mathbf{x};q,t),\text{ when
}\eta_{i}>\eta_{i+1},$ (3.6) $\displaystyle T_{i}f_{\eta}(\mathbf{x};q,t)$
$\displaystyle=tf_{\eta}(\mathbf{x};q,t),\text{ when }\eta_{i}=\eta_{i+1},$
(3.7) $\displaystyle q^{\eta_{n}}f_{\eta}(\mathbf{x};q,t)$
$\displaystyle=f_{\eta_{n},\eta_{1},\dots,\eta_{n-1}}(qx_{n},x_{1},\dots,x_{n-1};q,t).$

We say that a family of polynomials is a _KZ family_ if they satisfy the above
relations at $q=1$.

######  Remark 3.7.

Note that (3.7) can be rephrased as

$q^{\eta_{n}}f_{\eta}(\mathbf{x};q,t)=(\omega
f_{\eta_{n},\eta_{1},\dots,\eta_{n-1}})(\mathbf{x};q,t).$

######  Remark 3.8.

Using the fact that $T_{i}^{2}=(t-1)T_{i}+t$, together with (3.5), we see that
any qKZ family also satisfies

(3.8) $T_{i}f_{\eta}=(t-1)f_{\eta}+tf_{s_{i}\eta}\text{ when
}\eta_{i}<\eta_{i+1}.$

The following polynomials were first introduced in [CdGW15]. They were
subsequently shown to be generating functions for multiline queues in [CMW22],
see 5.5. They were called _ASEP polynomials_ by Chen, de Gier and Wheeler
[CdGW20].

######  Definition 3.9 (ASEP polynomials).

Given a partition $\lambda$, the _ASEP polynomials_

$\\{{F}_{\eta}:={F}_{\eta}(\mathbf{x};q,t)\ |\ \eta\in S_{\lambda}\\}$

are the unique family of polynomials which are a qKZ family and such that
${F}_{\lambda}(\mathbf{x};q,t)=E_{\lambda}(\mathbf{x};q,t).$

We can use the ASEP polynomials to define Macdonald polynomials. The fact that
3.10 agrees with the original definition of Macdonald polynomials comes from
[CdGW15, Lemma 1].

######  Definition 3.10.

Let $\lambda$ be a partition. We define the _Macdonald polynomial_
$P_{\lambda}(\mathbf{x};q,t)$ by

(3.9) $P_{\lambda}(\mathbf{x};q,t)=\sum_{\eta}{F}_{\eta}(\mathbf{x};q,t),$

where the sum runs over all $\eta\in S_{\lambda}$, i.e. the permutations
$\eta$ of $\lambda$.

When we specialize the $x_{i}$’s and $q$ to be $1$, we obtain a relation
between the ASEP polynomials and the multispecies ASEP.

######  Proposition 3.11.

[CdGW15, Corollary 1] Let $\lambda=(\lambda_{1},\dots,\lambda_{n})$ be a
partition. The steady state probability that the multispecies ASEP is in state
$\eta\in S_{\lambda}$ is

$\frac{{F}_{\eta}(1,\dots,1;1,t)}{P_{\lambda}(1,\dots,1;1,t)}.$

Macdonald showed [Mac95, Section VI, Chapter 4] that

(3.10) $P_{\lambda}(\mathbf{x};1,t)=e_{\lambda^{\prime}}(\mathbf{x}),$

where $\lambda^{\prime}$ denotes the conjugate partition of $\lambda$, and
$e_{\lambda^{\prime}}(\mathbf{x})$ denotes the corresponding elementary
symmetric polynomial.

Note that by (3.10), $P_{\lambda}(1,\dots,1;1,t)$ is independent of $t$.

######  Example 3.12.

Consider all permutations of the tuple $(0,1,2)$ as compositions. The ASEP
polynomials for each of them are as follows:

$\begin{array}[]{|c|c|}\hline\cr\eta&{F}_{\eta}(x_{1},x_{2},x_{3};q,t)\\\
\hline\cr(0,1,2)&x_{2}x_{3}^{2}+\frac{t(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
(0,2,1)&x_{2}^{2}x_{3}+\frac{(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
(1,0,2)&x_{1}x_{3}^{2}+\frac{(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
(1,2,0)&x_{1}x_{2}^{2}+\frac{qt(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
(2,0,1)&x_{1}^{2}x_{3}+\frac{qt(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
(2,1,0)&x_{1}^{2}x_{2}+\frac{q(1-t)}{1-qt^{2}}x_{1}x_{2}x_{3}\\\
\hline\cr\end{array}$

One can check that ${F}_{(2,1,0)}=E_{(2,1,0)}$ from 3.5.

######  Example 3.13.

The expansions of the Macdonald polynomials for partitions of size $3$ in the
variables $x_{1},x_{2},x_{3}$ in terms of monomial symmetric functions
$m_{\lambda}=m_{\lambda}(x_{1},x_{2},x_{3})$ are as follows:

$\begin{array}[]{|c|c|}\hline\cr\lambda&P_{\lambda}(x_{1},x_{2},x_{3};q,t)\\\
\hline\cr(3,0,0)&m_{(3)}+\frac{\left(q^{2}+q+1\right)(t-1)}{q^{2}t-1}m_{(2,1)}+\frac{(q+1)\left(q^{2}+q+1\right)(t-1)^{2}}{(qt-1)\left(q^{2}t-1\right)}m_{(1,1,1)}\\\\[5.69046pt]
\hline\cr(2,1,0)&m_{(2,1)}+\frac{(t-1)(2qt+q+t+2)}{qt^{2}-1}m_{(1,1,1)}\\\\[5.69046pt]
\hline\cr(1,1,1)&m_{(1,1,1)}\\\ \hline\cr\end{array}$

Note that the sum of the ASEP polynomials from 3.12 equals $P_{(2,1,0)}$ as
given above. It turns out that the expansion of the Macdonald polynomials is
the same irrespective of the number of variables (as long as there are enough
variables).

The following result will be useful in proving 1.4.

######  Lemma 3.14.

[CMW22, Theorem 3.4 (18)] Let $\eta$ be a composition and $1\leq i\leq n$.
Then ${F}_{\eta}+{F}_{s_{i}\eta}$ is symmetric in $x_{i}$ and $x_{i+1}$.

###  3.3. Permuted basement Macdonald polynomials

There is a more general class of polynomials that generalize both the
nonsymmetric Macdonald polynomials and the ASEP polynomials. These polynomials
are called _permuted basement Macdonald polynomials_
$E_{\alpha}^{\sigma}({\mathbf{x}};q,t)$ (where $\sigma\in S_{n}$ and $\alpha$
is a composition with $n$ parts); they were introduced by Ferreira in [Fer11]
and further studied in [Ale19] and [AS19]. We do not need here the definition,
but only the fact that they specialize to both the nonsymmetric Macdonald
polynomials and the ASEP polynomials. In particular, the nonsymmetric
Macdonald polynomial $E_{\eta}$ is equal to
$E_{\operatorname{rev}(\eta)}^{w_{0}}$, where $\operatorname{rev}(\eta)$
denotes the reverse composition $(\eta_{n},\eta_{n-1},\dots,\eta_{1})$ of
$\eta=(\eta_{1},\dots,\eta_{n})$ and $w_{0}$ denotes the longest permutation
$(n,\ldots,2,1)$ in one-line notation. We also have the following.

######  Proposition 3.15.

[CMW22, Proposition 4.1] For $\eta=(\eta_{1},\ldots,\eta_{n})$, define
$\operatorname{inc}(\eta)$ to be the sorting of the parts of $\eta$ in
increasing order. Then

$F_{\eta}=E_{\operatorname{inc}(\eta)}^{\sigma}$

where $\sigma$ is the element of $S_{n}$ with longest length such that

$\eta_{\sigma(1)}\leq\eta_{\sigma(2)}\leq\dots\leq\eta_{\sigma(n)}.$

##  4\. Nonsymmetric Macdonald and ASEP polynomials at $q=1$

The main goal of this section is to prove 4.18, which will be a main
ingredient in our proof of 1.1. Along the way we will prove various properties
of nonsymmetric Macdonald polynomials and ASEP polynomials at the
specialization $q=1$.

###  4.1. Nonsymmetric Macdonald polynomials at $q=1$

In this section we show that certain ratios of nonsymmetic Macdonald
polynomials at $q=1$ are elementary symmetric polynomials, building on work of
[AS19]. Our first main result is the following.

######  Theorem 4.1.

Consider a composition $\eta=(\eta_{1},\dots,\eta_{n})$ such that its parts of
sizes $a$ and $b$ (with $a<b$) occur in increasing order from left to right in
$(\eta_{1},\dots,\eta_{n})$, and such that $\eta$ has no parts of size $h$ for
any $a<h<b$. (In this case, we say that the classes $a$ and $b$ are
_adjacent_.) Let $\hat{\eta}$ be the composition obtained from $\eta$ by
changing all $a$’s to $b$, or changing all $b$’s to $a$. Then at $q=1$,

(4.1) $\frac{E_{\eta}}{E_{\hat{\eta}}}=\left(\frac{e_{\\#\\{i|\eta_{i}\geq
b\\}}}{e_{\\#\\{i|\hat{\eta}_{i}\geq b\\}}}\right)^{b-a}.$

######  Example 4.2.

At $q=1$, we have

$\displaystyle\frac{E_{2515}}{E_{2212}}$
$\displaystyle=\left(\frac{e_{2}}{e_{0}}\right)^{3},\qquad$
$\displaystyle\frac{E_{2515}}{E_{5515}}$
$\displaystyle=\left(\frac{e_{2}}{e_{3}}\right)^{3},$
$\displaystyle\frac{E_{201}}{E_{200}}$
$\displaystyle=\left(\frac{e_{2}}{e_{1}}\right)^{1},\qquad$
$\displaystyle\frac{E_{3311220}}{E_{3311110}}$
$\displaystyle=\left(\frac{e_{4}}{e_{2}}\right)^{1}.$

######  Remark 4.3.

Some results from [AS19] will be useful to us. However, we have to be careful
of conventions. Sage and [CMW22] have the same conventions, but those
conventions are different from [AS19]; in particular, the compositions
indexing Macdonald polynomials are reversed in these references. That is, the
polynomial called $E_{(\eta_{1},\dots,\eta_{n})}$ in [AS19] is the same as the
polynomial called $E_{(\eta_{n},\dots,\eta_{1})}$ in Sage and [CMW22].

######  Definition 4.4.

The _weak standardization_ of a composition $\eta$, denoted $\tilde{\eta}$, is
the lexicographically smallest composition with the property that if
$\eta_{i}<\eta_{j}$, then $\tilde{\eta}_{i}<\tilde{\eta}_{j}$ for all pairs
$i,j$. The _conjugate_ $\eta^{\prime}$ of a composition
$(\eta_{1},\dots,\eta_{n})$ is obtained by drawing a left-justified diagram
consisting of rows of lengths $(\eta_{1},\dots,\eta_{n})$, then reading the
columns from left to right and recording the number of boxes in each column.

For example, if $\eta=(2,5,1,4,4,5,4)$, then $\tilde{\eta}=(1,3,0,2,2,3,2)$,
and $\eta^{\prime}=(7,6,5,5,2).$ The theorem below is due to [AS19], but we
have phrased it using the conventions of Sage and [CMW22].

######  Theorem 4.5.

[AS19, Equation (2) and Theorem 18] Choose a basement $\sigma\in S_{n}$ and
composition $\eta=(\eta_{1},\dots,\eta_{n})$, and let $\tau=\tilde{\eta}$
denote the weak standardization of $\eta$. Then

(4.2)
$E_{\eta}^{\sigma}(\mathbf{x};1,t)=\left(\frac{e_{\eta^{\prime}}(\mathbf{x})}{e_{\tau^{\prime}}(\mathbf{x})}\right)E_{{\tau}}^{\sigma}(\mathbf{x};1,t),$

where $\eta^{\prime}$ denotes the composition which is conjugate to $\eta$,
and $e_{\eta^{\prime}}(\mathbf{x})/e_{\tau^{\prime}}(\mathbf{x})$ is an
elementary symmetric polynomial independent of $t$. And if the composition
$\eta$ has weakly increasing parts, then

$E_{\eta}^{\sigma}(\mathbf{x};1,t)=e_{\eta^{\prime}}(\mathbf{x}).$

Note that if $\mu$ is a partition and $\eta\in S_{\mu}$, then
$\eta^{\prime}\in S_{\mu^{\prime}}$, that is, both $\eta^{\prime}$ and
$\mu^{\prime}$ have the same set of parts. Using 4.5 and 3.15, we now obtain
the following.

######  Corollary 4.6.

Let $\mu=(\mu_{1},\dots,\mu_{n})$ be a partition, and let
$\eta=(\eta_{1},\dots,\eta_{n})\in S_{\mu}$. Then we have

$\frac{{F}_{\eta}(\mathbf{x};1,t)}{{F}_{\tilde{\eta}}(\mathbf{x};1,t)}=h_{\mu}(\mathbf{x}),$

where
$h_{\mu}(\mathbf{x}):=e_{\mu^{\prime}}(\mathbf{x})/e_{\tilde{\mu}^{\prime}}(\mathbf{x})$
is a (symmetric) polynomial in $\mathbf{x}=(x_{1},\dots,x_{n})$ which is
independent of $\eta$ and of $t$.

And if the composition $\eta$ has weakly increasing parts, then the
nonsymmetric Macdonald polynomial $E_{\eta}$ satisfies

$E_{\eta}(\mathbf{x};1,t)=e_{\eta^{\prime}}(\mathbf{x}).$

######  Proposition 4.7.

[AS19, Proposition 16] If $\eta$ is a composition, then

$\frac{E_{\eta}(\mathbf{x};1,t)}{E_{\tilde{\eta}}(\mathbf{x};1,t)}$

is an elementary symmetric polynomial.

Before proving 4.1, we recall the _shape permuting operator_ from [HHL08,
Equation (17)].

######  Proposition 4.8.

[HHL08] Let $\nu$ be a composition, and suppose $\nu_{i}>\nu_{i+1}$. Write

$r_{i}(\nu)=\\#\\{j<i\mid\nu_{i+1}<\nu_{j}\leq\nu_{i}\\}+\\#\\{j>i\mid\nu_{i+1}\leq\nu_{j}<\nu_{i}\\}.$

Then

(4.3)
$E_{s_{i}\nu}(\mathbf{x};q,t)=\left(T_{i}+\frac{1-t}{1-q^{\nu_{i+1}-\nu_{i}}t^{r_{i}(\nu)}}\right)E_{\nu}(\mathbf{x};q,t).$

(In the terminology of [HHL08], the exponent $\nu_{i+1}-\nu_{i}$ of $q$
appearing in Equation 4.3 is $1+{\operatorname{leg}}(u)$, and the exponent
$r_{i}(\nu)$ of $t$ is ${\operatorname{arm}}(u)$, where $u$ is the box
$(i,\nu_{i+1}+1)$ in the column diagram of the composition $\nu$.)

###### Proof of 4.1.

We start by reducing the proof of 4.1 to the case of a composition whose parts
are weakly increasing. Note that by our assumptions on $\eta$ and
$\hat{\eta}$, we will have $\eta_{i}>\eta_{i+1}$ if and only if
$\hat{\eta}_{i}>\hat{\eta}_{i+1}$. In this case, it is not hard to see that
$r_{i}(\eta)=r_{i}(\hat{\eta})$. Thus if we set $q=1$, then the shape
permuting operator is the “same” for both $\eta$ and $\hat{\eta}$, that is,

$E_{s_{i}\eta}(\mathbf{x};1,t)=\left(T_{i}+\frac{1-t}{1-t^{r}}\right)E_{\eta}(\mathbf{x};1,t)$

and

$E_{s_{i}\hat{\eta}}(\mathbf{x};1,t)=\left(T_{i}+\frac{1-t}{1-t^{r}}\right)E_{\hat{\eta}}(\mathbf{x};1,t),$

where $r=r_{i}(\eta)=r_{i}(\hat{\eta})$. By applying the same sequence of
shape permuting operators at $q=1$ to $E_{\eta}(\mathbf{x};1,t)$ and
$\left(\frac{e_{\\#\\{i|\eta_{i}\geq b\\}}}{e_{\\#\\{i|\hat{\eta}_{i}\geq
b\\}}}\right)^{b-a}E_{\hat{\eta}}(\mathbf{x};1,t)$, and using 3.1, we can
therefore reduce to the case of a composition whose parts are weakly
increasing.

We now suppose that $\eta$ has weakly increasing parts; likewise $\hat{\eta}$
has weakly increasing parts. We now apply 4.6, which says that at $q=1$,

$\frac{E_{\eta}}{E_{\hat{\eta}}}=\frac{e_{\eta^{\prime}}}{e_{\hat{\eta}^{\prime}}}.$

We have that
$\frac{e_{\eta^{\prime}}}{e_{\hat{\eta}^{\prime}}}=\left(\frac{e_{\\#\\{i|\eta_{i}\geq
b\\}}}{e_{\\#\\{i|\hat{\eta}_{i}\geq b\\}}}\right)^{b-a},$ so we are done. ∎

Now what we are really interested in is quantities likes $E_{201}/E_{100}$ or
$E_{3311220}/E_{2211110}$. The denominator is obtained in two stages from the
numerator: by merging two adjacent classes $a<b$ appearing in increasing order
as in 4.1, and then applying weak standardization, as in 4.7. By applying 4.1
and 4.7, we obtain the following.

######  Corollary 4.9.

Consider a composition $\eta=(\eta_{1},\dots,\eta_{n})$ such that its parts of
sizes $a$ and $b$ (with $a<b$) occur in increasing order from left to right in
$(\eta_{1},\dots,\eta_{n})$, and such that $\eta$ has no parts of size $h$ for
any $a<h<b$. Let $\hat{\eta}$ be the composition obtained from $\eta$ by
changing all $a$’s to $b$, or changing all $b$’s to $a$. Let
$\tilde{\hat{\eta}}$ be the composition obtained from $\hat{\eta}$ by applying
the weak standardization (i.e. we find the smallest composition whose parts
are in the same relative order as those of $\hat{\eta}$). Then at $q=1$,
$E_{\eta}/E_{\tilde{\hat{\eta}}}$ is a ratio of elementary symmetric
polynomials.

###  4.2. Projections of ASEP polynomials at $q=1$

In this section we explain some properties of ASEP polynomials that hold true
at $q=1$. These properties are algebraic analogues of the recolouring
properties of the $t$-PushTASEP discussed in Section 2.1, in particular 2.3.

Recall the definition of weakly order-preserving from 2.1. Throughout this
section we will fix the following notation.

######  Notation 4.10.

Let $\phi:\mathbb{N}\to\mathbb{N}$ be a weakly order-preserving function. Let
$\mu=(\mu_{1},\dots,\mu_{n})$ and $\lambda=(\lambda_{1},\dots,\lambda_{n})$ be
partitions with $\phi(\lambda)=\mu$. For $\eta\in S_{\mu}$, let

(4.4) $G_{\eta}=G_{\eta}(\mathbf{x};t):=\sum_{\begin{subarray}{c}\zeta\in
S_{\lambda}\\\ \phi(\zeta)=\eta\end{subarray}}F_{\zeta}(\mathbf{x};1,t).$

######  Example 4.11.

Suppose that $\phi(0)=\phi(1)=1$ and $\phi(j)=j$ for $j\notin\\{0,1\\}$. Let
$\lambda=(2,1,0)$ and $\mu=\phi(\lambda)=(2,1,1)$. Then

$\displaystyle G_{(2,1,1)}$
$\displaystyle=F_{(2,1,0)}(\mathbf{x};1,t)+F_{(2,0,1)}(\mathbf{x};1,t)=x_{1}(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}),$
$\displaystyle G_{(1,2,1)}$
$\displaystyle=F_{(1,2,0)}(\mathbf{x};1,t)+F_{(0,2,1)}(\mathbf{x};1,t)=x_{2}(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}),$
$\displaystyle G_{(1,1,2)}$
$\displaystyle=F_{(1,0,2)}(\mathbf{x};1,t)+F_{(0,1,2)}(\mathbf{x};1,t)=x_{3}(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}),$

where the values of the ASEP polynomials are taken from 1.2.

######  Proposition 4.12.

Use 4.10. Then the family $\\{G_{\eta}\ |\ \eta\in S_{\mu}\\}$ is a KZ family,
i.e. it satisfies the relations of 3.6 at $q=1$.

###### Proof.

It suffices to prove 4.12 when $\phi$ has a particular form: namely, there is
a natural number $\ell$ such that $\phi(\ell)=\phi(\ell+1)=\ell+1$, and
$\phi(j)=j$ for $j\notin\\{\ell,\ell+1\\}$. Suppose that the components of
$\lambda$ contain $a$ instances of $\ell$ and $b$ instances of $\ell+1$. Then
each $\eta\in S_{\mu}$ contains $a+b$ instances of $\ell+1$, and $G_{\eta}$ is
a sum of the ${a+b\choose a}$ polynomials $F_{\zeta}$, where $\zeta$ is
obtained from $\eta$ by changing $a$ of the $\ell+1$’s in $\eta$ to $\ell$.

Suppose that $1\leq i\leq n-1$ is a position such that at most one of
$\eta_{i}$ and $\eta_{i+1}$ equals $\ell+1$. Then each $\zeta$ occurring in
the right-hand-side of (4.4) has the property that $\zeta_{i}$ and
$\zeta_{i+1}$ have the same relative order as $\eta_{i}$ and $\eta_{i+1}$.
Without loss of generality say that $\eta_{i}>\eta_{i+1}$. (The arguments in
the other cases are similar.) Then we have that

$\displaystyle T_{i}G_{\eta}$ $\displaystyle=\sum_{\begin{subarray}{c}\zeta\in
S_{\lambda}\\\
\phi(\zeta)=\eta\end{subarray}}T_{i}(F_{\zeta}(\mathbf{x};1,t))$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta\in S_{\lambda}\\\
\phi(\zeta)=\eta\end{subarray}}F_{s_{i}(\zeta)}(\mathbf{x};1,t)$
$\displaystyle=G_{s_{i}(\eta)},$

as desired, where we used the fact (cf 3.9) that the polynomials
$F_{\zeta}(\mathbf{x};q,t)$ are themselves a qKZ family.

Now suppose that $\eta_{i}=\eta_{i+1}=\ell+1$. Then the $\zeta$ appearing on
the right-hand side of (4.4) satisfy either $\zeta_{i}=\zeta_{i+1}$, or
$\zeta_{i}=\ell+1$ and $\zeta_{i+1}=\ell$, or $\zeta_{i}=\ell$ and
$\zeta_{i+1}=\ell+1$, so we will divide up the sum accordingly. In what
follows, we will abbreviate $F_{\zeta}(\mathbf{x};1,t)$ by $F_{\zeta}$, and
omit the conditions that $\zeta\in S_{\lambda}$ and $\phi(\zeta)=\eta$ in all
sums below. So we get

$\displaystyle T_{i}G_{\eta}$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta_{i}=\zeta_{i+1}\end{subarray}}T_{i}(F_{\zeta})+\sum_{\begin{subarray}{c}\zeta_{i}=\ell+1,\zeta_{i+1}=\ell\end{subarray}}T_{i}(F_{\zeta})+\sum_{\begin{subarray}{c}\zeta_{i}=\ell,\zeta_{i+1}=\ell+1\end{subarray}}T_{i}(F_{\zeta}).$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta_{i}=\zeta_{i+1}\end{subarray}}tF_{\zeta}+\sum_{\begin{subarray}{c}\zeta_{i}=\ell,\zeta_{i+1}=\ell+1\end{subarray}}F_{\zeta}+\sum_{\begin{subarray}{c}\zeta_{i}=\ell,\zeta_{i+1}=\ell+1\end{subarray}}\left((t-1)F_{\zeta}+tF_{s_{i}\zeta}\right).$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta_{i}=\zeta_{i+1}\end{subarray}}tF_{\zeta}+\sum_{\begin{subarray}{c}\zeta_{i}=\ell,\zeta_{i+1}=\ell+1\end{subarray}}\left(tF_{\zeta}+tF_{s_{i}\zeta}\right).$
$\displaystyle=tG_{\eta},$

as desired.

Finally since the polynomials $F_{\zeta}$ satisfy (3.7), at $q=1$ we get

$F_{\zeta}(\mathbf{x};1,t)=F_{(\zeta_{n},\zeta_{1},\dots,\zeta_{n-1})}(x_{n},x_{1},\dots,x_{n-1};1,t).$

But then we have

$\displaystyle G_{\eta}(\mathbf{x};t)$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta\in S_{\lambda}\\\
\phi(\zeta)=\eta\end{subarray}}F_{\zeta}(\mathbf{x};1,t)$
$\displaystyle=\sum_{\begin{subarray}{c}\zeta\in S_{\lambda}\\\
\phi(\zeta)=\eta\end{subarray}}F_{(\zeta_{n},\zeta_{1},\dots,\zeta_{n-1})}(x_{n},x_{1},\dots,x_{n-1};1,t)$
$\displaystyle=G_{(\eta_{n},\eta_{1},\dots,\eta_{n-1})}(x_{n},x_{1},\dots,x_{n-1},t),$

as desired. ∎

######  Lemma 4.13.

Suppose that $\\{f_{\eta}\ |\ \eta\in S_{\lambda}\\}$ and $\\{g_{\eta}\ |\
\eta\in S_{\lambda}\\}$ are both KZ families. If the ratio
$h:=g_{\lambda}/f_{\lambda}$ of the partition-indexed terms in the two
families is symmetric in $x_{1},\dots,x_{n}$, then the families $(f_{\eta})$
and $(g_{\eta})$ are proportional to each other, i.e. $g_{\eta}/f_{\eta}=h$
for all $\eta\in S_{\lambda}$.

###### Proof.

Let $\lambda$ be a partition and let $\eta\in S_{\lambda}$. We can obtain
$\eta$ from $\lambda$ by a sequence of nearest-neighbour transpositions, each
of which changes a pair of entries from decreasing order into increasing
order. Hence from (3.5) we have $f_{\eta}=T_{i_{k}}\dots T_{i_{1}}f_{\lambda}$
and $g_{\eta}=T_{i_{k}}\dots T_{i_{1}}g_{\lambda}$, for some sequence
$i_{1},\dots i_{k}$. Then by induction and 3.1, if $g_{\lambda}=hf_{\lambda}$
where $h$ is symmetric in $x_{1},\dots,x_{n}$, then $g_{\eta}=hf_{\eta}$. ∎

######  Definition 4.14.

Use 4.10. Let $\operatorname{inc}(\lambda,\phi)$ be the lexicographically
smallest $\zeta\in S_{\lambda}$ such that $\phi(\zeta)=\mu$.

######  Example 4.15.

Suppose $\phi(6)=\phi(5)=\phi(4)=5,\phi(3)=4,\phi(2)=\phi(1)=1$. Let
$\lambda=(6,6,6,5,4,3,3,2,2,1,1)$, so that $\mu=(5,5,5,5,5,4,4,1,1,1,1)$. Then
$\operatorname{inc}(\lambda,\phi)=(4,5,6,6,6,3,3,1,1,2,2)$.

######  Proposition 4.16.

Use 4.10. We have that $G_{\mu}=E_{\operatorname{inc}(\lambda,\phi)}(q=1)$.

###### Proof.

The ASEP polynomials and the nonsymmetric Macdonald polynomials are related
via a triangular change of basis [CdGW15, (23)], and hence the span of
$\\{E_{\eta}(q=1)\ |\ \eta\in S_{\lambda}\\}$ is the same as the span of
$\\{F_{\eta}(q=1)\ |\ \eta\in S_{\lambda}\\}$. Therefore $G_{\mu}$ lies in the
span of $\\{E_{\eta}(q=1)\ |\ \eta\in S_{\lambda}\\}$.

We know from 4.12 that $\\{G_{\eta}\ |\ \eta\in S_{\mu}\\}$ is a KZ family.
Proceeding as in the proof of [CMW22, Lemma 1.23], we can use the relations of
the KZ family to show that $G_{\mu}$ is an eigenvector of each $Y_{i}$, and
that the eigenvalue of $Y_{i}$ on $G_{\mu}$ is the same as the eigenvalue of
$Y_{i}$ on $F_{\mu}=E_{\mu}$ when $q=1$. But now it is easy to see from (3.4)
that the eigenvalues of $Y_{i}$ on $E_{\mu}$ at $q=1$ are the same as the
eigenvalues of $Y_{i}$ on $E_{\operatorname{inc}(\lambda,\phi)}$ at $q=1$.
Recall from 3.4 that the polynomials $\\{E_{\eta}(q=1),\eta\in S_{\lambda}$}
have distinct tuples of eigenvalues at $q=1$. Since $G_{\mu}$ is in the span
of those polynomials and its eigenvalues agree with those of
$E_{\operatorname{inc}(\lambda,\phi)}$, it must be a multiple of
$E_{\operatorname{inc}(\lambda,\phi)}$. Since both polynomials have the same
leading term, it follows that in fact
$G_{\mu}=E_{\operatorname{inc}(\lambda,\phi)}$. ∎

######  Theorem 4.17.

Use 4.10. We have that at $q=1$, $E_{\operatorname{inc}(\lambda,\phi)}$ is a
symmetric function multiple of $E_{\mu}=F_{\mu}$.

###### Proof.

Since $\mu$ is a partition, we have that $F_{\mu}=E_{\mu}$ from the definition
of ASEP polynomials. Now this theorem can be obtained by repeated applications
of 4.9, together with 4.7. ∎

######  Theorem 4.18.

Use 4.10. For all $\eta\in S_{\mu}$ and for $q=1$, we have

$\frac{G_{\eta}}{P_{\lambda}}=\frac{G_{\eta}}{\sum_{\theta\in
S_{\mu}}G_{\theta}}=\frac{F_{\eta}}{\sum_{\theta\in
S_{\mu}}F_{\theta}}=\frac{F_{\eta}}{P_{\mu}}.$

###### Proof.

By 4.16, we have that $G_{\mu}=E_{\operatorname{inc}(\lambda,\phi)}(q=1)$.
Since $\mu$ is a partition, we also know by 3.9 that $F_{\mu}=E_{\mu}$. So by
4.17, $h:=G_{\mu}/F_{\mu}(q=1)$ is symmetric in $x_{1},\dots,x_{n}$. We can
therefore apply 4.13 to conclude that $G_{\eta}=hF_{\eta}$ for all $\eta\in
S_{\mu}$. But now it follows that

$\frac{G_{\eta}}{\sum_{\theta\in
S_{\mu}}G_{\theta}}=\frac{F_{\eta}(q=1)}{\sum_{\theta\in
S_{\mu}}F_{\theta}(q=1)}.$

The other equalities in the proposition follow after using the fact that
$\sum_{\theta\in S_{\mu}}F_{\theta}=P_{\mu}$ and $\sum_{\theta\in
S_{\mu}}G_{\theta}=\sum_{\xi\in S_{\lambda}}F_{\xi}(q=1)=P_{\lambda}(q=1).$ ∎

######  Example 4.19.

We continue 4.11. Let $\eta=(1,2,1)$. Then by 4.18, when we specialize to
$q=1$, we have

$\displaystyle\frac{G_{(1,2,1)}}{P_{(2,1,0)}}$
$\displaystyle=\frac{G_{(1,2,1)}}{G_{(2,1,1)}+G_{(1,2,1)}+G_{(1,1,2)}}$
$\displaystyle=\frac{F_{(0,2,1)}+F_{(1,2,0)}}{F_{(2,0,1)}+F_{(2,1,0)}+F_{(0,2,1)}+F_{(1,2,0)}+F_{(0,1,2)}+F_{(1,0,2)}}$
$\displaystyle=\frac{F_{(1,2,1)}}{F_{(2,1,1)}+F_{(1,2,1)}+F_{(1,1,2)}}=\frac{F_{(1,2,1)}}{P_{(2,1,1)}}{=\frac{x_{2}}{x_{1}+x_{2}+x_{3}}}.$

##  5\. Multiline diagrams

In this section we define _multiline diagrams_. These are combinatorial
objects which were introduced in the context of the multispecies TASEP in
[FM09], and have subsequently been generalised to a range of related settings,
including in [Mar20] in the context of the multispecies ASEP, and in [CMW22]
in the context of Macdonald polynomials.

###  5.1. Definition of multiline diagrams

The definition we present here is a special case of the definition given in
[CMW22]: we impose $q=1$, and only define multiline diagrams with bottom row
$\eta$, where $\eta=(\eta_{1},\dots,\eta_{n})$ is a composition whose nonzero
parts are distinct.

Fix a partition $\lambda=(\lambda_{1},\dots,\lambda_{n})=\langle
s^{m_{s}},\dots,1^{m_{1}},0^{m_{0}}\rangle$, where $\lambda_{1}=s$ and
$m_{i}\in\\{0,1\\}$ for all $i\geq 1$. Let $a_{i}=\sum_{r=i}^{s}m_{r}$ be the
number of particles of type $i$ or higher.

######  Definition 5.1.

A _ball system_ with _content $\lambda$_ is an array with $s$ rows and $n$
columns in which each of the $sn$ positions is either empty or occupied by a
ball, and in which the number of balls in row $i$ is $a_{i}$. We number the
rows from $1$ to $s$ from bottom to top, and the sites from $1$ to $n$ from
left to right.

######  Definition 5.2.

Given a ball system $B$ with content $\lambda$, a _multiline diagram_ $D$
(with pattern $B$) is an assignment of types (positive integers) to the balls
of $B$. The $a_{i}$ balls in row $i$ are given distinct labels from the set
(of size $a_{i}$) $\\{r\geq i:m_{r}=1\\}$. The labelling satisfies the
following constraint: if two vertically adjacent sites $(i,j)$ and $(i,j+1)$
both contain a ball, then the label of the lower site $(i,j)$ must be at least
as large as the label of the upper site $(i,j+1)$. We also call $\lambda$ the
content of the diagram.

Each row $i$ of a multiline diagram $D$ gives rise to a composition
$\rho^{(i)}=(\rho^{(i)}_{1},\dots,\rho^{(i)}_{n})$, where

$\rho^{(i)}_{j}=\begin{cases}h&\text{if $D$ has a ball with label $h$ at
$(i,j)$};\\\ 0&\text{if $(i,j)$ is empty in $D$}.\end{cases}$

The composition $\rho^{(1)}$ associated to the bottom row of $D$ is sometimes
called the _type_ of $D$. If $D$ has content $\lambda$ then $\rho^{(1)}(D)\in
S_{\lambda}$.

###  5.2. Generating multiline diagrams

We next explain a procedure for randomly generating a multiline diagram with
content $\lambda$. As above let $s$ be the largest entry of $\lambda$, so that
the diagram has $s$ rows.

We first generate the ball system. The occupancies on different rows are
independent. On row $r$, where we require $a_{r}$ balls, we occupy a given set
of sites $A_{r}\subset[n]$ of size $a_{r}$ with probability proportional to
$\prod_{j\in A_{r}}x_{j}$. (This is precisely the stationary distribution of
the single-type $t$-PushTASEP, as given in 2.4 – the normalising constant is
given by the elementary symmetric function $e_{a_{r}}(x_{1},\dots,x_{n})$.)

Now we assign labels to the balls in the system. This is done recursively line
by line, working from top (row $s$) to bottom (row $1$).

  * •

We assign the single ball on row $s$ (the top row) the label $s$.

  * •

Now suppose we have already labelled the rows from $s$ down to $r+1$, and it
is time to label row $r$. We consider the balls in row $r+1$ one by one in
decreasing order of their label. We match each one to a ball on row $r$, and
that ball on row $r$ will be given the same label.

Suppose we are considering the ball with label $h$ on row $r+1$, with position
in some column $j$. First we check whether the site immediately below it,
$(r,j)$, has a ball which has not yet been labelled. If so we match to that
ball, labelling it $h$. This is called a trivial match. Otherwise, consider
all the balls remaining in row $r$ which have not yet been labelled – there
are a total of $K:=a_{r}-a_{h+1}$ of them. Suppose their columns, listed from
left to right in cyclic order starting from column $a$, are
$j_{1},j_{2},\dots,j_{K}$: that is,

$0<(j_{1}-j)\pmod{n}<(j_{2}-j)\pmod{n}<\dots<(j_{K}-j)\pmod{n}.$

Now we match the $h$-labelled ball at $(r+1,j)$ to the ball at position
$(r,j_{k})$ with probability $t^{k-1}/(1+t+\dots+t^{K-1})$, and assign the
label $h$ to that ball.

  * •

In this way every ball on row $r+1$ gets matched to a ball on row $r$. If
$a_{r}=a_{r+1}$ then we have labelled every ball on row $r$, and we move on to
labelling the balls in the rows below. If instead $a_{r}=a_{r+1}+1$, then
there remains a single unlabelled ball on row $r$, and we assign it label $r$.

Proceeding in this way we construct a labelling having the properties in 5.2.

###  5.3. Weight function for multiline diagrams

Closely related to the above sampling procedure, we define a weight function
on multiline diagrams. For a given collection of particle counts, and given
parameters $(x_{1},\dots,x_{n})$ and $t$, the probability of sampling a given
diagram using the procedure above is proportional to its _weight_ , as defined
in 5.3.

######  Definition 5.3.

Let $D$ be a multiline diagram with pattern $B$, where $B$ is a $s\times n$
ball system.

For $1\leq j\leq n$ let $c_{j}$ be the number of balls in column $j$. Then the
$x$-weight of the diagram is defined by
$\operatorname{wt}_{x}(D)=\prod_{j=1}^{n}x_{j}^{c_{j}}$.

The $t$-weight is defined as follows. Consider $h\geq 2$ such that $m_{h}=1$.
Then there is one ball with label $h$ in each of the rows $h$ and below. For
each $r=1,\dots,h-1$ we associate a local weight $w_{D}(h,r)$ to the ball with
label $h$ in row $r$ as follows:

  * •

If the balls of label $h$ in rows $r+1$ and $r$ are in the same column
(corresponding to a trivial match), then $w_{D}(h,r)=1$.

  * •

Otherwise:

    * –

Let $K$ be the number of balls in row $r$ with label at most $h$;

    * –

Let $j$ be the column with the ball labelled $h$ in row $r+1$, and
$j^{\prime}$ the column with the ball labelled $h$ in row $r$. Consider the
interval with left endpoint $j$ and right endpoint $j^{\prime}$ (wrapping
cyclically around the ring if necessary). Let $\ell$ be the number of balls in
row $r$ between columns $j$ and $j^{\prime}$ with label less than $h$. We have
$0\leq\ell\leq K-1$.

Then define

(5.1) $w_{D}(h,r)=\frac{t^{\ell}}{1+t+\dots+t^{K-1}}.$

The $t$-weight of the diagram is then the product of all these $w_{D}(h,r)$
weights:

(5.2)
$\operatorname{wt}_{t}(D)=\prod_{r=1}^{s-1}\prod_{\begin{subarray}{c}r<h\leq
s:\\\ m_{h}=1\end{subarray}}w_{D}(h,r).$

Finally we define the _weight_ $\operatorname{wt}(D)$ of diagram $D$ to be the
product of its $x$-weight and _$t$ -weight_, that is,

$\operatorname{wt}(D)=\operatorname{wt}_{x}(D)\operatorname{wt}_{t}(D).$

Note first that $\operatorname{wt}_{x}(D)$ is proportional to the probability
of generating the ball system $B$ of $D$ in the first step of the procedure
above. (The constant of proportionality is
$\prod_{r=1}^{s}e_{a_{r}}(x_{1},\dots,x_{n})$.)

Also note that for each $h$ and $r$, $w_{D}(h,r)$ is precisely the probability
of making the given matching of the $h$-labelled ball between rows $r+1$ and
$r$ at the relevant step of the labelling process.

As a result, the conditional probability of obtaining the configuration of $D$
on row $r$, given the ball system and the configuration of $D$ on rows $s$
down to $r+1$, is the product

(5.3) $\prod_{\begin{subarray}{c}h:r<h\leq s:\\\
m_{h}=1\end{subarray}}w_{D}(h,r),$

which depends on $D$ only through its rows $r$ and $r+1$.

Hence the probability of generating a given diagram $D$ is proportional to
$\operatorname{wt}_{x}(D)$ multiplied by the product of (5.3) over $r$ from
$s-1$ down to $1$. This yields exactly $\operatorname{wt}(D)$, as required.
See Figure 3 for an example of a multiline diagram and its weight function.

5555544443331 Figure 3. A multiline diagram $D$ with $n=6$ columns and $s=5$
rows, with content $\lambda=(5,4,3,1,0,0)$ and bottom row
$\rho^{(1)}(D)=(4,0,1,5,3,0)\in S_{\lambda}$. It has weight
$\operatorname{wt}(D)=\operatorname{wt}_{x}(D)\operatorname{wt}_{t}(D)=x_{1}^{3}x_{3}^{2}x_{4}^{4}x_{5}^{2}x_{6}^{2}\,t^{2}$.

######  Lemma 5.4.

Let $\lambda=(\lambda_{1},\dots,\lambda_{n})=\langle
s^{m_{s}},\dots,1^{m_{1}},0^{m_{0}}\rangle$ be a partition with distinct
entries, no entry equal to $1$, and exactly one entry equal to $0$. That is,
$m_{i}\in\\{0,1\\}$, $m_{1}=0$, and $m_{0}=1$. Let $\eta,\eta^{\prime}\in
S_{\lambda}$, and $j\in\\{1,\dots,n\\}$. Then the following quantities are
equal.

  * •

the probability in the $t$-PushTASEP of transitioning from state $\eta$ to
state $\eta^{\prime}$, when a bell rings at site $j$ in state $\eta$;

  * •

given a multiline diagram with content $\lambda$, such that row $2$ has
configuration $\eta$ and the unique vacancy in row $1$ is at site $j$, the
conditional probability that row $1$ has configuration $\eta^{\prime}$.

###### Proof.

Note that by the condition on $\lambda$, both row $2$ and row $1$ of the
diagram have a single vacant site. It follows from the constraint in 5.2 that
the first quantity in 5.4 is nonzero if and only if the second quantity in 5.4
is nonzero.

From (5.3) with $r=1$, the conditional probability of obtaining a specific
configuration $\eta^{\prime}$ on row $1$, given the configuration $\eta$ on
row $2$, is given by

$\prod_{\begin{subarray}{c}h:1<h\leq s:\\\ m_{h}=1\end{subarray}}w_{D}(h,1),$

where $D$ is any diagram agreeing with $\eta$ and $\eta^{\prime}$ on rows $2$
and $1$ respectively.

But because of the equivalence of (2.2) and (5.1), that conditional
probability is exactly the same as (2.2), which is the probability of
obtaining $\eta^{\prime}$ when a bell rings at site $j$ in the state $\eta$
under the $t$-PushTASEP dynamics.

In the special case where $\eta_{j}=\eta^{\prime}_{j}=0$, i.e. both rows $1$
and $2$ of the multiline diagram $D$ have their unique vacancy in position
$j$, all particles in $D$ must form a trivial match between rows $2$ and $1$,
and the configuration in the two rows is identical. Correspondingly, under the
$t$-PushTASEP dynamics, if the bell rings at the site of an existing vacancy,
then the system stays in its current state. ∎

###  5.4. ASEP polynomials from multiline diagrams

The following result, which is a special case of a result from [CMW22],
relates the distribution of the bottom row of a multiline diagram with bottom
row $\eta$ to the ASEP polynomial indexed by $\eta$.

######  Theorem 5.5 ([CMW22, Definition 1.9, Theorem 1.25, Lemma 1.26]).

For any composition $\eta=(\eta_{1},\dots,\eta_{n})$ whose nonzero parts are
distinct, the ASEP polynomial $F_{\eta}(\mathbf{x};1,t)$ at $q=1$ can be
computed in terms of multiline diagrams as follows:

(5.4)
$F_{\eta}(\mathbf{x};1,t)=\sum_{D:\,\rho^{(1)}(D)=\eta}\operatorname{wt}(D)$

######  Remark 5.6.

The result from [CMW22] is more general than 5.5 because it holds for any
composition and for general $q$. However, we do not need the more general
version in this paper.

######  Remark 5.7.

Let $\lambda=(\lambda_{1},\dots,\lambda_{n})=\langle
s^{m_{s}},\dots,0^{m_{0}}\rangle$. The ASEP polynomials appearing as
numerators of stationary probabilities in 1.1 are polynomials in $\mathbf{x}$
with coefficients which are rational functions (but not necessarily
polynomials) in $t$. Using the connection with multiline diagrams, one can
show that these probabilities can be rewritten with numerators that are
polynomials in both $\mathbf{x}$ and $t$ and with common denominator given by

(5.5)
$P_{\lambda}(\mathbf{x};1,t)\prod_{i=1}^{s}\frac{[m_{1}+\cdots+m_{i}]_{t}!}{[m_{i}]_{t}!}=e_{\lambda^{\prime}}(\mathbf{x})\prod_{i=1}^{s}\frac{[m_{1}+\cdots+m_{i}]_{t}!}{[m_{i}]_{t}!}.$

In the case of 1.2, this common denominator is

$(t+1)\left(x_{1}+x_{2}+x_{3}\right)\left(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}\right).$

The factor in (5.5) involving $t$-factorials is the same as that in [Mar20]
for the ASEP.

##  6\. Proof of 1.1

In this section we prove 1.1. We start by proving it in the case where
$\lambda$ has distinct parts, and then we generalize it to the case of
repeated parts, using properties of ASEP polynomials at $q=1$.

###  6.1. Proof of 1.1 when $\lambda$ has distinct parts

######  Lemma 6.1.

Let $\lambda$ be a partition with distinct entries and no entry equal to $1$.
Consider a random multiline diagram $D$ with content $\lambda$, with
distribution proportional to the weight $\operatorname{wt}(D)$. The
configurations given by the bottom row (row $1$) and by the next-to-bottom row
(row $2$) have the same distribution.

###### Proof.

Let $\phi$ be the weakly order-preserving function given by $\phi(x)=x-1$ for
all $x\geq 1$ and $\phi(0)=0$. If $s$ is the largest entry of $\lambda$, then
$\phi(\lambda)$ has largest entry $s-1$.

Let $D$ be a random multiline diagram with content $\lambda$, with
distribution proportional to weight. The diagram $D$ has $s$ rows. Recall that
$\rho^{(1)}$ denotes the sequence of balls in the bottom row of $D$, and
$\rho^{(2)}$ denotes the sequence of balls in row $2$ of $D$.

Let $D^{\prime}$ be a random multiline diagram with content $\phi(\lambda)$,
again distributed proportional to weight. The diagram $D^{\prime}$ has $s-1$
rows.

In view of the generation process from Section 5.2, if we take rows $s$ down
to $2$ of the diagram $D$, and subtract $1$ from the label of every ball, then
the resulting diagram has distribution identical to that of the diagram
$D^{\prime}$. In particular, comparing row $2$ of $D$ to row $1$ of
$D^{\prime}$, we have that $\phi(\rho^{(2)}(D))$ and $\rho^{(1)}(D^{\prime})$
have the same distribution.

But we may instead compare row $1$ of $D$ to row $1$ of $D^{\prime}$. By 5.5,
for $\eta\in S_{\lambda}$, the probability that $\rho^{(1)}(D)=\eta$ equals
$\frac{{F}_{\eta}(\mathbf{x};1,t)}{\sum_{\tau\in
S_{\lambda}}{F}_{\tau}(\mathbf{x};1,t)}$, and the probability that
$\rho^{(1)}(D^{\prime})=\phi(\eta)$ equals
$\frac{{F}_{\phi(\eta)}(\mathbf{x};1,t)}{\sum_{\nu\in
S_{\phi(\lambda)}}{F}_{\nu}(\mathbf{x};1,t)}$. But $\eta$ and $\phi(\eta)$
have the same weak standardisation, so by 4.6, these two probabilities are
equal. It follows that $\phi(\rho^{(1)}(D))$ has the same distribution as
$\rho^{(1)}(D^{\prime})$.

We have proved that both $\phi(\rho^{(1)}(D))$ and $\phi(\rho^{(2)}(D))$ have
the same distribution as $\rho^{(1)}(D^{\prime})$. But $\phi$ is a bijection
from $S_{\lambda}$ to $S_{\phi(\lambda)}$. So in fact $\rho^{(1)}(D)$ and
$\rho^{(2)}(D)$ have the same distribution, as required. See Figure 4 for an
illustration. ∎

444433322333221 Figure 4.  On the left, a multiline diagram $D$ with content
$\lambda=(4,3,2,0)$. On the right, a multiline diagram $D$ with content
$\phi(\lambda)=(3,2,1,0)$ (where $\phi$ is defined as in the proof of 6.1).
The configurations $\rho^{(2)}(D)$ and $\rho^{(1)}(D)$ (the two lowest rows of
$D$) have the same distribution, and the distribution of $\phi(\rho^{(1)}(D))$
and of $\phi(\rho^{(2)}(D))$ is the same as that of $\rho^{(1)}(D^{\prime})$.

Write $p_{j}(\eta,\eta^{\prime})$ for the probability of obtaining
$\eta^{\prime}$ from $\eta$ using the $t$-PushTASEP jump dynamics when a bell
rings at site $j$.

Now we average over $j$. Specifically, we take a weighted average of $p_{j}$
with weights proportional to $x_{j}^{-1}$:

(6.1)
$p(\eta,\eta^{\prime})=\sum_{j}\frac{x_{j}^{-1}}{x_{1}^{-1}+x_{2}^{-1}+\dots+x_{n}^{-1}}p_{j}(\eta,\eta^{\prime}).$

Since $x_{j}^{-1}$ is the rate at which the bell at site $j$ in the
$t$-PushTASEP, we have that $p$ defined by (6.1) gives the transition
probabilities of a discrete-time Markov chain whose stationary distribution is
the same as that of the continuous-time $t$-PushTASEP.

By 5.4, $p_{j}(\eta,\eta^{\prime})$ also gives the probability of obtaining
$\eta^{\prime}$ as row $1$ of a multiline diagram with content $\lambda$,
given that row $2$ is $\eta$ and that the vacancy in the bottom row is at site
$j$. But the probability of having a vacancy at $j$ is proportional to
$x_{j}^{-1}$, independently of the rows above, so $p(\eta,\eta^{\prime})$
defined by (6.1) is the probability of obtaining $\eta^{\prime}$ as row $1$ of
the diagram, given that row $2$ is $\eta$.

We also know from 6.1 that the distributions of row $1$ and row $2$ are the
same. Since row $1$ is obtained from row $2$ by a single update of the
dynamics $p$, this tells us that their common distribution is a stationary
distribution for $p$, and hence also for the continuous-time $t$-PushTASEP.

But this common distribution is also proportional to the ASEP polynomials. So
we obtain that $\pi_{\lambda}(\eta),\eta\in S_{\lambda}$ is proportional to
$F_{\eta}$ as required to give the result of 1.1 in this case.

###  6.2. Proof of 1.1 in the general case

To extend the result from the previous section to prove the general case of
1.1, we apply the results from Section 4.

Consider some partition $\mu$ (as ever, assumed to have at least one entry
$0$). We can find some $\lambda$ satisfying the conditions of Section 6.1
(i.e. having distinct entries, no entry $1$, and exactly one entry $0$) and
some weakly order-preserving function $\phi$, such that $\mu=\phi(\lambda)$.
2.3 then tells us that

(6.2) $\pi_{\mu}(\eta)=\sum_{\zeta\in
S_{\lambda}:\phi(\zeta)=\eta}\pi_{\lambda}(\zeta).$

But we also know from 1.1 (which we now know holds in the case that $\lambda$
has distinct parts) that for $\zeta\in S_{\lambda}$,

$\pi_{\lambda}(\zeta)=\frac{{F}_{\zeta}(\mathbf{x};1,t)}{P_{\lambda}(\mathbf{x};1,t)}.$

We then obtain from (6.2) that

$\pi_{\mu}(\eta)=\sum_{\begin{subarray}{c}\zeta\in S_{\lambda}\\\
\phi(\zeta)=\eta\end{subarray}}\frac{{F}_{\zeta}(\mathbf{x};1,t)}{P_{\lambda}(\mathbf{x};1,t)}.$

Now by (4.4) we have that

$\pi_{\mu}(\eta)=\frac{G_{\eta}(\mathbf{x},t)}{P_{\lambda}(\mathbf{x};1,t)}.$

Finally using 4.18, we obtain that

$\pi_{\mu}(\eta)=\frac{F_{\eta}(\mathbf{x},t)}{P_{\mu}(\mathbf{x};1,t)},$

as desired. This completes the proof of 1.1.

###### Proof of 1.4.

Let $A\subset S_{\lambda}$ be any event (i.e. collection of configurations)
that is conserved by exchanging the contents of sites $i$ and $i+1$; that is,
$A=s_{i}A=\\{s_{i}\eta\mid\eta\in A\\}$.

Since $s_{i}$ is a bijection on $S_{\lambda}$, we then have $\sum_{\eta\in
A}F_{\eta}=\sum_{\eta\in A}F_{s_{i}\eta}$, giving

$\sum_{\eta\in A}F_{\eta}=\frac{1}{2}\sum_{\eta\in
A}\left(F_{\eta}+F_{s_{i}\eta}\right).$

By 3.14, this quantity is symmetric in $x_{i}$ and $x_{i+1}$.

Now suppose $A$ depends only on the configuration in sites $1,2,\dots,k$. Then
the above holds for any $i$ with $k<i<n$. We obtain that for any such $i$, the
probability of $A$ in the stationary distribution,

$p^{(\lambda)}(A):=\sum_{\eta\in A}p^{(\lambda)}_{\eta}=\sum_{\eta\in
A}\frac{F_{\eta}(x_{1},\dots,x_{n};1,t)}{P_{\lambda}(\mathbf{x};1,t)},$

is symmetric in $x_{i}$ and $x_{i+1}$ (since the Macdonald polynomial in the
denominator is symmetric). But then the probability is in fact symmetric in
all of $x_{k+1},\dots,x_{n}$. This gives the symmetry required for 1.4. ∎

##  7\. Formulas for density and currents

In this section we discuss the density of particles and the current in the
$t$-PushTASEP. In particular, we give a formula for the density of particles
in 7.1. We also prove 1.5, which gives a formula for the current in the single
species case. We end with a discussion of the current in the multispecies
case.

Let us consider the single species $t$-PushTASEP first. By 2.4, the density
(in the stationary distribution) of particles is the same as for the PushTASEP
at $t=0$. As we will explain, this continues to hold for the multispecies
$t$-PushTASEP. Let $\eta^{(i)}_{j}$ denote the occupation variable for the
particle of species $i$ at site $j$, i.e $\eta^{(i)}_{j}=1$ (resp.
$\eta^{(i)}_{j}=0$) provided the $j$’th site is occupied (resp. not occupied)
by $i$. The formula for the density of particles in the multispecies PushTASEP
is obtained directly from that in the single species case: the density of the
particle of species $j$ is the density of the particle of species $1$ in the
single species PushTASEP with $m_{j}+\cdots+m_{s}$ particles minus the density
of the particle of species $1$ in the single species PushTASEP with
$m_{j+1}+\cdots+m_{s}$ particles. The formula for the density, shown in 7.1
below, is the same as given for the $t=0$ case in [AM23, Proposition 18]. The
proof is identical and is omitted.

######  Corollary 7.1.

The density of species $r$ at the first site in the multispecies $t$-PushTASEP
with content $\lambda=\langle s^{m_{s}},\dots,0^{m_{0}}\rangle$ on
$n=\sum_{i}m_{i}$ sites is given by

$\langle\eta^{(r)}_{1}\rangle=x_{1}\frac{s_{\langle
2^{a_{r+1}},1^{m_{r}-1}\rangle}(x_{2},\dots,x_{n})}{e_{a_{r}}(\mathbf{x})e_{a_{r+1}}(\mathbf{x})},$

where $s_{\mu}$ is the Schur polynomial indexed by $\mu$, and
$a_{r}=m_{r}+\cdots+m_{s}$ for $1\leq r\leq s$.

We now move on to studying the current (at stationarity) for the single
species $t$-PushTASEP. Let $\lambda=\langle 1^{m_{1}},0^{m_{0}}\rangle$, where
$m_{1}+m_{0}=n$, so that we are studying a system with $m_{1}$ particles and
$m_{0}$ vacancies. Recall that the _current (at stationarity)_ of a particle
across a given edge (say $(n,1)$) of the lattice is the number of particles
per unit time that cross that edge in the long-time limit. Because of particle
conservation, the current is independent of the edge. We will denote the
stationary current in our system of particles by $J_{m_{0},m_{1}}$.

In terms of the stationary distribution for the $t$-PushTASEP, we can compute
$J_{m_{0},m_{1}}$ as follows. If a particle is at position $k$ and there is a
vacancy at position $j<k$, then the particle at $k$ can make a transition to
$j$, and contributes to the current across the edge $(n,1)$ in doing so. If
there are $h$ vacancies in sites $k+1,\dots,n,1,\dots,j-1$, then the rate of
this transition is $t^{h}/x_{k}$. Formally, we can write the current as

(7.1)
$J_{m_{0},m_{1}}=\sum_{h=0}^{m_{0}-1}\sum_{j=1}^{n-m_{0}+h}\sum_{k=j+1}^{n}\frac{t^{h}}{x_{k}}\sum_{\eta}\pi(\eta),$

where the sum on the right is over all

$\\{\eta\in S_{\langle 1^{m_{1}},0^{m_{0}}\rangle}\ |\
\eta_{k}=1,\eta_{j}=0,\text{ and }\eta\text{ has $h$ vacancies in sites
$k+1,\dots,n,1\dots j-1$}\\}.$

The formula we need to prove 1.5 is the following identity for elementary
symmetric functions, which seems to be new.

######  Lemma 7.2.

Fix $n$ and $m_{0}<n$ positive integers. Then, for all $0\leq h\leq m_{0}-1$,
we have

(7.2) $(h+1)e_{n-m_{0}-1}(x_{1},\dots,x_{n})=\\\
\sum_{a=m_{0}-h}^{n-1}\sum_{j=1}^{n-a}e_{h-m_{0}+a}(x_{j+1},\dots,x_{j+a-1})\times
e_{n-h-1-a}(x_{j+a+1},\dots,x_{n},x_{1},\dots,x_{j-1}).$

###### Proof.

For convenience, set $m_{1}=n-m_{0}$. We need to show that every monomial in
$e_{m_{1}-1}$ occurs on the right hand side exactly $h+1$ times. So fix a
subset $S=\\{1\leq s_{1}<\cdots<s_{m_{1}-1}\leq n\\}$ and consider the summand
in the right hand side of (7.2). It depends on two parameters $j$ and $a$ and
we set $k=j+a$. Thus, the sum depends on two parameters, $j$ and $k$ instead.
The first factor in the summand is an elementary symmetric function depending
on the variables strictly between $j$ and $k$, and the second is one depending
on variables between $k$ and $j$ counted cyclically. Therefore, we must choose
$j$ and $k$ to be in the set $[n]\setminus S$ such that there are $h+k-j-
m_{0}$ elements of $S$ between $j$ and $k$. To complete the proof, it will
suffice to show that there are exactly $h+1$ many choices.

To make the argument easier to follow, let us first consider the case where
$h=m_{0}-1$. Then, we have to choose $j$ and $k$ so that there is no element
of $[n]\setminus S$ strictly between them. There is exactly one way of
choosing $j\in[1,s_{1}-2]$ and that is with $k=j+1$. We can also choose
$j=s_{1}-1$ with $k=s_{1}+1$. Thus, $j$ can be chosen to be any position
between $1$ and $s_{1}-1$ in exactly one way. Similarly, $j$ can be chosen to
be any position between $s_{i}+1$ and $s_{i+1}-1$ in one way, for $1\leq i\leq
m_{1}-2$. Lastly, $j$ can be chosen to be any element between $s_{m_{1}-1}+1$
and $n-1$ with $k=j+1$. Summing all of these possibilities, we get
$n-m_{1}=m_{0}=h+1$, which is independent of $S$.

The argument for general $h$ goes the same way. Between each $s_{i}+1$ and
$s_{i+1}-1$, there is exactly one way of choosing $k$ so that there are
$h+k-j-m_{0}$ elements of $S$ between $j$ and $k$, for small values of $i$.
The change occurs at the end as we get closer to $n$. Every time the value of
$h$ increases by $1$, the number of possibilities of $j$ reduces exactly by
$1$. It is easy to see that this argument is independent of the choice of $S$.

As a sanity check, consider the case of $h=0$. In that case, the only
possibility is to choose $j$ (resp. $k$) to be the smallest (resp. largest)
element of $[n]\setminus S$, which is consistent with what we want to prove. ∎

###### Proof of 1.5.

The current is independent of the edge being considered. So look at the edge
connecting $n$ to $1$. For a particle hop to count towards the current across
this edge, it must hop from a site $k\in[2,n]$ to a vacant site $j<k$. If
there are $h$ holes between the sites $k+1$ to $j-1$ (of which there are
$n-k+j-1$ many), then the rate of this transition is $t^{h}/([m_{0}]x_{k})$.
Therefore the total stationary weight of these configurations is
$e_{n-h-1-k+j}(x_{k+1},\dots,x_{n},x_{1},\dots,x_{j-1})$. Similarly, there are
$m_{0}-h-1$ holes between the $k-j-1$ sites between $j+1$ and $k-1$ and so the
total stationary weight of such configurations is
$e_{h-m_{0}+k-j}(x_{j+1},\dots,x_{j+a-1})$. Summing over all possible values
of $j$ and $k$, we see that the current is

$J_{m_{0},m_{1}}=\sum_{h=0}^{m_{0}-1}\sum_{j=1}^{n-m_{0}+h}\sum_{k=j+1}^{n}\frac{t^{h}}{[m_{0}]}\frac{e_{h-m_{0}+k-j}(x_{j+1},\dots,x_{j+a-1})}{e_{m_{1}}(x_{1},\dots,x_{n})}\\\
\times e_{n-h-1-k+j}(x_{k+1},\dots,x_{n},x_{1},\dots,x_{j-1}).$

Now substituting $a=j-k$ and using 7.2, we arrive at

$J_{m_{0},m_{1}}=\frac{e_{m_{1}-1}(x_{1},\dots,x_{n})}{e_{m_{1}}(x_{1},\dots,x_{n})}\sum_{h=0}^{m_{0}-1}\frac{(h+1)t^{h}}{[m_{0}]},$

which gives the desired result. ∎

Now we would like to compute the current for the multispecies case. The
current of species $j$ in the $t$-PushTASEP on $\Omega_{\lambda}$ is the
difference of the total currents of species $j$ through $s$ minus the total
currents of species $j+1$ through $s$. Following the argument in [AM23,
Proposition 19], we would like to calculate both these in terms of the single
species $t$-PushTASEP using 1.5 and 2.2. Unfortunately, this does not work as
in the $t=0$ case if $j<s$. The main reason is that an edge can contribute
towards multiple currents in a single transition when $t>0$.

We illustrate this with the example of $\lambda=(2,1,0)$ shown in Figure 2.
Consider the current of species $1$ across the edge $(3,1)$. Using the
colouring argument, we would get this current to be

$J_{1,2}-J_{2,1}=\frac{e_{1}(x_{1},x_{2},x_{3})}{e_{1}(x_{1},x_{2},x_{3})}=\frac{1+2t}{1+t}\frac{e_{0}(x_{1},x_{2},x_{3})}{e_{1}(x_{1},x_{2},x_{3})},$

which gives, after some manipulations,

$J_{1,2}-J_{2,1}=\frac{(1+t)(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}{(1+t)e_{2,1}(x_{1},x_{2},x_{3})}.$

Now let us calculate the current by brute force. Particle $1$ jumps across the
edge $(3,1)$ only for the following states when either the $1$ jumps, or when
the $2$ jumps displacing the $1$:

  * •

$(0,1,2)$,

  * •

$(0,2,1)$,

  * •

$(2,0,1)$.

The sum of these contributions gives

$\frac{x_{2}x_{3}}{e_{2,1}(x_{1},x_{2},x_{3})}\left(\frac{x_{1}t}{1+t}+x_{3}\right)\left(\frac{1}{x_{2}}+\frac{t}{x_{3}(1+t)}\right)\\\
+\frac{x_{2}x_{3}}{e_{2,1}(x_{1},x_{2},x_{3})}\left(\frac{x_{1}}{1+t}+x_{2}\right)\left(\frac{1}{x_{3}}+\frac{1}{x_{2}(1+t)}\right)\\\
+\frac{x_{1}x_{3}}{e_{2,1}(x_{1},x_{2},x_{3})}\left(x_{1}+\frac{x_{2}t}{1+t}\right)\left(\frac{1}{x_{3}}+\frac{t}{x_{1}(1+t)}\right),$

which after simplifying becomes

$\frac{(1+t)^{2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+(1+2t+2t^{2})(x_{1}x_{2}+x_{1}x_{3}+x_{2},x_{3})}{(1+t)^{2}e_{2,1}(x_{1},x_{2},x_{3})},$

and this does not match $J_{1,2}-J_{2,1}$. The main reason is that (i) in the
transition from $(0,1,2)$ where the $2$ displaces the $1$, both particles end
up crossing the edge $(3,1)$.

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