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\title[Refined Littlewood identity for spin Hall--Littlewood functions]{Refined Littlewood identity for spin Hall--Littlewood symmetric rational functions}
\author[\initial{S.} Gavrilova]{\firstname{Svetlana} \lastname{Gavrilova}}
\address{National Research University Higher School of Economics\\ Faculty of Mathematics\\ 6 Usacheva st.\\ Moscow\\ 119048 (Russia)}
\email{sveta\_6117@mail.ru}
\thanks{The author is partially supported by International Laboratory of Cluster Geometry NRU HSE, RF Government grant, ag. \textnumero 075-15-2021-608.}
\keywords{Littlewood identity, symmetric functions, six vertex model}
\subjclass{05E05, 82B20, 81R50}
\datereceived{2021-07-07}
\daterevised{2022-02-24}
\dateaccepted{2022-05-29}
\begin{document}
\begin{abstract}
Fully inhomogeneous spin Hall--Littlewood symmetric rational
functions $F_\lambda$ are multiparameter deformations of the classical Hall--Littlewood symmetric polynomials and can be viewed as partition functions in $\mathfrak{sl}(2)$ higher spin six vertex models.
We obtain a refined Littlewood identity
expressing a
weighted sum of $F_\lambda$'s over all signatures $\lambda$ with even multiplicities
as a certain Pfaffian. This Pfaffian can be derived as
a partition function of the six vertex model in a triangle
with suitably decorated domain wall boundary conditions.
The proof
is based on the Yang--Baxter equation.
\end{abstract}
\maketitle
\noindent
\section{Introduction}
\label{sec:intro}
\subsection{Background}
In the present paper we deal with summation
identities for spin Hall--Little\-wood symmetric
rational functions. These functions
arise
as
partition functions
of square lattice integrable
vertex models
related to the quantum group $U_q(\widehat{\mathfrak{sl}_2})$. This description originally appeared in \cite{Bor17,BP18}.
The spin Hall--Littlewood functions
also can be identified with Bethe Ansatz eigenfunctions
of the higher spin six vertex model on $\mathbb{Z}$, \cf \cite[Ch. VII]{KBI93}.
They also appear as eigenfunctions of certain stochastic particle systems
\cite{Pov13,BCPS15,CP16}. Following
\cite{Bor17,BP18} and subsequent works, we treat spin Hall--Littlewood functions and their relatives from the point of
view of the theory of symmetric functions.
A classical reference on the theory of symmetric functions is the book
\cite{Mac95} where Schur, Hall--Littlewood, and Macdonald symmetric
polynomials
and symmetric functions are developed and various identities for them are formulated or proved.
One of the common features
for most
families of symmetric polynomials
is a \emph{Littlewood type summation identity}. For example, the Schur symmetric polynomials $s_{\lambda}$ satisfy the following Littlewood identity:
\begin{equation*}
\sum_{\lambda'\ \text{even}} s_{\lambda}(u_1, \dots, u_m)
=
\prod_{1 \leq i