\documentclass[ALCO,ThmDefs,Unicode,epreuves]{cedram}
\OneNumberAllTheorems
\definecolor{OliveGreen}{rgb}{0,0.6,0}
%\usepackage{pifont}
%\usepackage{amsmath,amssymb,commath}
\usepackage{commath}
%\usepackage[utf8]{inputenc}
%\usepackage[T1]{fontenc}
\usepackage[capitalize,noabbrev]{cleveref}
\Crefname{theo}{Theorem}{Theorems}
\Crefname{prop}{Proposition}{Propositions}
\Crefname{lemm}{Lemma}{Lemmas}
\Crefname{remm}{Remark}{Remarks}
\Crefname{exam}{Example}{Examples}
\Crefname{coro}{Corollary}{Corollaries}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,calc}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes}
\usepackage{tikz-cd}
\usepackage{mathscinet}
\usepackage{enumitem}
\usepackage{array,multirow}
\hyphenation{tab-leaux}
\newcommand{\PreserveBackslash}[1]{\let\temp=\\#1\let\\=\temp}
\let\PBS=\PreserveBackslash
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Cbb}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\KO}{\mathbb{A}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\Gcal}{\mathcal{G}}
\newcommand{\Sym}[1]{\mathfrak{S}_{#1}}
\newcommand{\Spin}[1]{\widetilde{\mathfrak{S}}_{#1}}
\newcommand{\SGA}[1]{\Cbb \Sym{#1}^-}
\newcommand{\ChSpin}{\Ch^{\textup{spin}}}
\newcommand{\ChD}{\widetilde{\Ch}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\irrepSp}{\psi}
\newcommand{\double}{D^*}
\newcommand{\Dover}{D_{\textup{over}}}
\newcommand{\class}{\pi}
\newcommand{\partition}{I}
\newcommand{\sett}{S}
\newcommand{\setJ}{T}
\newcommand{\const}{C}
\newcommand{\CHIL}[2]{\chi^{#1}\left(#2\right)}
\newcommand{\XX}[2]{X^{#1}\left(#2\right)}
\newcommand{\PHI}[2]{\widetilde{\phi}^{#1}\left(#2\right)}
\newcommand{\PHIeasy}[2]{\phi^{#1}\left(#2\right)}
\newcommand{\proP}{\mathbf{p}}
\newcommand{\proQ}{\mathbf{q}}
\newcommand{\stimes}{\boxtimes}
\DeclareRobustCommand{\stirlingS}{\genfrac\{\}{0pt}{}}
\DeclareRobustCommand{\stirlingF}{\genfrac[]{0pt}{}}
\newcommand{\Part}{\mathcal{P}}
\newcommand{\SP}{\mathcal{SP}}
\newcommand{\OP}{\mathcal{OP}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\ttop}{top}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\dec}{dec}
\DeclareMathOperator{\Ch}{Ch}
\DeclareMathOperator{\spanrm}{span}
%%%%%%%%% a placer avant \begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}
\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Auteur
\author{\firstname{Sho} \lastname{Matsumoto}}
\address{ Graduate School of Science and Engineering \\ Kagoshima University \\
1-21-35 \\ Korimoto \\ Kagoshima \\ Japan
}
\email{shom@sci.kagoshima-u.ac.jp}
\author{\firstname{Piotr} \lastname{{\'S}niady}}
\address{Institute of Mathematics \\ Polish Academy of Sciences \\
ul.~{\'S}niadeckich 8\\ 00-956 Warszawa \\ Poland
}
\email{psniady@impan.pl}
\thanks{Research of Sho Matsumoto was supported by JSPS KAKENHI Grant Number 17K05281.
Research of Piotr {\'S}niady was supported by \emph{Narodowe Centrum Nauki}, grant number
2017/26/A/ST1/00189.}
%%%%% Sujet
\subjclass{20C25, 20C30, 05E05}
\keywords{Projective representations of the symmetric groups, linear
representations of the symmetric groups, asymptotic representation theory, Stanley character formula}
%%%%% Gestion
%\DOI{10.5802/alco.92}
%\datereceived{2018-11-23}
%\daterevised{2019-07-08}
%\dateaccepted{2019-08-01}
%%%%% Titre et résumé
\title[Linear versus spin]{\tralicstex{Linear versus spin:
representation theory of the symmetric groups}{Linear versus spin:
representation theory of~the symmetric groups}}
\begin{abstract}\
We relate the \emph{linear} asymptotic representation theory of the symmetric
groups to its \emph{spin} counterpart. In particular, we give explicit formulas
which express the normalized irreducible \emph{spin} characters evaluated on a
strict partition $\xi$ with analogous normalized \emph{linear} characters
evaluated on the double partition $D(\xi)$. We also relate some natural
filtration on the usual (\emph{linear}) Kerov--Olshanski algebra of polynomial
functions on the set of Young diagrams with its \emph{spin} counterpart.
Finally, we give a spin counterpart to Stanley formula for the characters of the
symmetric groups in terms of counting maps.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
\emph{Spin} representation theory of the symmetric groups is a younger sibling
of the usual \emph{linear} representation theory. In the past, development of
the spin sibling took a path \emph{parallel} to that of the linear sibling
(often with a decades long delay): the spin versions of the notions and
relationships were modelled after their linear analogues but there were somewhat
distinct and they had to be developed from scratch. The paths of the
developments of the siblings did not really \emph{cross}: there were not many
results which would directly link these two realms.
In the current paper we consider a setup in which there is a direct link between
the spin and the linear representation theory of the symmetric groups. In fact
we claim that \emph{the spin representation theory} (or at least some of its
aspects) \emph{can be derived from its linear counterpart}.
Before we state our results
we shall recall some basic concepts of the spin representation theory of the
symmetric groups. For more details and bibliographic references refer to~\cite{Wan2012,Kleshchev2005,Ivanov2004}.
\subsection{Spin representation theory of the symmetric groups}
\label{sec:projective-representations}
\subsubsection{Linear representations}
Recall that a \emph{linear} representation of a finite group $G$
is a group homomorphism $\irrepSp \colon G \to \GL(V)$
to the group of \emph{linear} transformations
$\GL(V)$
of some finite-dimensional complex vector space~$V$.
\subsubsection{Projective representations}
A \emph{projective} representation of a finite group $G$
is a group homomorphism $\irrepSp \colon G \to \PGL(V)$
to the group of \emph{projective linear} transformations
$\PGL(V)=\GL(V)/\Cbb^{\times} $ of the projective space $P(V)$
for some finite-dimensional complex vector space $V$.
Equivalently, a projective representation can be viewed as a
map $\phi\colon G \to \GL(V)$ to the general linear group with the property that
\[\phi(x) \phi(y)=c_{x,y}\ \phi(xy) \]
holds true for all $x,y\in G$ for some non-zero scalar $c_{x,y}\in \Cbb^{\times}$.
Each irreducible \emph{linear} representation $\irrepSp\colon G\to\GL(V)$ gives
rise to its projective version $\irrepSp\colon G\to\PGL(V)$. The irreducible
projective representations \emph{which cannot be obtained in this way} are
called \emph{irreducible spin representations} and are in the focus of the
current paper.
\subsubsection{Spin symmetric group and spin characters}
\label{sec:spin}
The \emph{spin group $\Spin{n}$}~\cite{Schur1911}
is a double cover of the symmetric group:
\begin{equation}
\label{eq:long-sequence}
1 \longrightarrow \Z_2=\{1,z\} \longrightarrow \Spin{n} \longrightarrow \Sym{n} \longrightarrow 1.
\end{equation}
More specifically, it is the
group generated by $t_1,\dots,t_{n-1},z$
subject to the relations:
\begin{align*}
z^2 &= 1 ,\\
z t_i &= t_i z, & t_i^2&= z & \text{for $i\in [n-1]$}, \\
(t_i t_{i+1})^3 &= z & & & \text{for $i\in [n-2]$}, \\
t_i t_j &= z t_j t_i & & & \text{for $|i-j|\geq 2$};
\end{align*}
we use the convention that $[k]=\{1,\dots,k\}$. Under the mapping
$\Spin{n}\to\Sym{n}$ the generators $t_1,\dots,t_{n-1}$ are mapped to the
Coxeter tranpositions $(1,2),\ (2,3),\dots,(n-1,n)\in\Sym{n}$.
The main advantage of the spin group comes from the fact that any \emph{projective} representation
$\irrepSp\colon \Sym{n}\to \PGL(V)$ of the \emph{symmetric group} can be lifted
to a \emph{linear} representation
$\widetilde{\irrepSp}\colon\Spin{n}\to\GL(V)$ of the \emph{spin group} so that
the following diagram commutes:
\[ \begin{tikzcd}
\Spin{n} \arrow[r, "\widetilde{\irrepSp}"] \arrow[d]
& \GL(V) \arrow[d] \\
\Sym{n} \arrow[r, "\irrepSp"]
& \PGL(V). \end{tikzcd}
\]
In this way the \emph{projective} representation theory of $\Sym{n}$ is
equivalent to the \emph{linear} representation theory of $\Spin{n}$ which allows
to speak about the characters.
The
irreducible \emph{spin} representations of $\Sym{n}$ turn out to correspond to
irreducible \emph{linear} representations of the \emph{spin group algebra}
$\SGA{n}:= \Cbb\Spin{n} / \langle z+1 \rangle$ which is the quotient of the group
algebra $\Cbb\Spin{n}$ by the ideal generated by $(z+1)$.
\subsubsection{Partitions}
An \emph{integer partition} of $n\geq 0$ (often referred to as \emph{Young
diagram} with $n$ boxes) is a finite sequence
$\lambda=(\lambda_1,\dots,\lambda_\ell)$ of positive integers (called
\emph{parts}) which is weakly decreasing $\lambda_1\geq \cdots\geq \lambda_\ell$
and such that $n=|\lambda|=\lambda_1+\cdots+\lambda_\ell$. The set of integer
partitions of a given integer $n\geq 0$ will be denoted by $\Part_n$, and the
set of \emph{all} integer partitions by $\Part=\bigsqcup_{n\geq 0} \Part_n$.
We sometimes write $\lambda \vdash n$ instead of $\lambda \in \Part_n$.
An integer partition is \emph{odd} if all of its parts are odd integers;
we denote by $\OP$ the set of odd partitions and
by $\OP_n$ the set of odd partitions of a given integer $n\geq 0$.
\looseness-1
An integer partition $\xi=(\xi_1,\dots,\xi_\ell)$ is \emph{strict}
if it has no repeated entries; in other words the corresponding sequence $\xi_1>\cdots>\xi_\ell$ is
\emph{strictly} decreasing. We denote by $\SP$
the set of strict partitions and by $\SP_n$ the set of strict partitions of a
given integer $n\geq 0$.
We visualize partitions as Young diagrams drawn in the French convention and
strict partitions as \emph{shifted Young diagrams}, cf.~\cref{fig:strict}.
\begin{figure}[htb]
\centerline{
\begin{tikzpicture}[xscale=0.6,yscale=0.6]
\begin{scope}
\clip (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0);
\draw[gray] (0,0) grid (8,3);
\end{scope}
\draw[->] (-1,0) -- (8,0) node[anchor=west]{$x$};
\draw[->] (-1,0) -- (-1,4) node[anchor=south]{$y$};
\draw[ultra thick] (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0) -- cycle;
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (0.5,0.5) -- (5.5,0.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (1.5,1.5) -- (5.5,1.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (2.5,2.5) -- (3.5,2.5);
\end{tikzpicture}
\hspace{3ex}
\begin{tikzpicture}[xscale=0.6,yscale=0.6]
\begin{scope}[xshift=1cm]
\clip (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0);
\draw[gray] (0,0) grid (8,3);
\end{scope}
\begin{scope}[rotate=90,yscale=-1]
\clip (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0);
\draw[gray] (0,0) grid (8,3);
\end{scope}
\begin{scope}[xshift=1cm]
\draw[ultra thick] (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0) -- cycle;
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (0.5,0.5) -- (5.5,0.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (1.5,1.5) -- (5.5,1.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (2.5,2.5) -- (3.5,2.5);
\end{scope}
\begin{scope}[rotate=90,yscale=-1]
\draw[ultra thick] (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0) -- cycle;
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (0.5,0.5) -- (5.5,0.5);
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (1.5,1.5) -- (5.5,1.5);
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (2.5,2.5) -- (3.5,2.5);
\end{scope}
\draw[line width=3.2pt, dashed,red] (0,0) -- (1,0) -- (1,1) -- (2,1) -- (2,2) -- (3,2) -- (3,3) -- (4,3) -- (4,4) -- (5,4) -- (5,5) -- (6,5) -- (6,6);
\end{tikzpicture}
}
\caption{ Strict partition $\xi=(6,5,2)$ shown as a \emph{shifted Young
diagram} and its double $D(\xi)=(7,7,5,3,2,2)$. }
\label{fig:strict}
\label{fig:double}
\end{figure}
\subsubsection{Spin characters}
\label{sec:intro-spin}
Schur~\cite{Schur1911} proved that, roughly
speaking\footnote{\label{note1}For a precise statement see~\cref{sec:spin-characters}.},
the conjugacy classes of $\Spin{n}$ which are non-trivial from the
viewpoint of the spin character theory are indexed by the elements of the set
$\OP_n$, \ie by the odd partitions of $n$.
Schur also proved that, roughly speaking\textsuperscript{(\ref{note1})}, the
irreducible spin representations of $\Spin{n}$ are indexed by the elements of
the set $\SP_n$, \ie by strict partitions of $n$
For an odd partition $\pi\in\OP_n$ (which corresponds to a conjugacy class of
$\Spin{n}$) and a strict partition $\xi\in\SP_n$ (which corresponds to its
irreducible spin representation) we denote by $\PHIeasy{\xi}{\pi}$ the corresponding
\emph{spin character} (for some fine details related to this definition we refer
to~\cref{sec:spin-characters}).
\subsection{Normalized characters}
The usual way of viewing the linear characters $\CHIL{\lambda}{\pi}$ of the
symmetric group $\Sym{n}$ is to fix the irreducible representation~$\lambda$ and
to consider the character as a function of the conjugacy class $\class$.
The \emph{dual approach}, initiated by Kerov and Olshanski~\cite{Kerov1994},
suggests to do the opposite: \emph{fix the conjugacy class $\class$ and to view
the character as a function of the irreducible representation $\lambda$}.
In
order for this approach to be successful one has to choose the most convenient
normalization constants which we review in the following.
Following Kerov and Olshanski~\cite{Kerov1994},
for a fixed integer partition $\class$ the
corresponding \emph{normalized linear character on the conjugacy class $\class$}
is the function on the set of \emph{all} Young diagrams given by
\begin{equation}
\label{eq:linear-characters}
\Ch_\class(\lambda):=\begin{cases}
n^{\downarrow k}
\ \frac{ \CHIL{\lambda}{\class\cup 1^{n-k}} }{ \CHIL{\lambda}{1^{n}} } &
\text{if } n\geq k, \\
0 & \text{otherwise,}
\end{cases}
\end{equation}
where $n=|\lambda|$ and $k=|\class|$ and $n^{\downarrow k}=n (n-1) \cdots
(n-k+1)$ denotes the falling power. Above, for partitions $\lambda,\sigma\vdash
n$ we denote by $\CHIL{\lambda}{\sigma}$ the irreducible linear character of the
symmetric group which corresponds to the Young diagram $\lambda$, evaluated on
any permutation with the cycle decomposition given by $\sigma$. For more context
we refer to the monographs~\cite[Section~7.3]{Meliot2017} and
\cite[Eq.~(2.21)]{Hora2016}.
Following Ivanov~\cite{Ivanov2004,Ivanov2006}, for a fixed odd partition
$\class\in\OP$ the corresponding \emph{normalized spin character} is the
function on the set of \emph{all} strict partitions given by
\begin{equation}
\label{eq:projective-normalized}
\ChSpin_\class(\xi):=\begin{cases}
n^{\downarrow k}\ 2^{\frac{k-\ell(\class)}{2}}
\ \frac{ \PHIeasy{\xi}{\class\cup 1^{n-k}}}{ \PHIeasy{\xi}{1^{n}}}
& \text{if } n\geq k, \\
0 & \text{otherwise,}
\end{cases}
\end{equation}
where $n=|\xi|$, $k=|\class|$,
and $\ell(\pi)$ denotes the number of parts of $\pi$.
In the following we shall discuss briefly some heuristics behind these definitions.
Both types of normalized characters~\eqref{eq:linear-characters} and
\eqref{eq:projective-normalized} involve evaluation of the irreducible (linear
or spin) character on the partition $\pi \cup 1^{n-k}$, \ie on the partition
$\pi$ augmented by the appropriate number of parts, all equal to $1$.
In the case of the linear characters this operation is motivated by considering
the infinite sequence of inclusions of the symmetric groups
\[ \Sym{1}\subseteq \Sym{2} \subseteq \Sym{3} \subseteq \cdots \]
which corresponds to viewing the symmetric group $\Sym{n}$ as the set of these
permutations of the set $\mathbb{N}=\{1,2,\dots\}$ which are equal to the
identity outside of $[n]$. Thanks to these inclusions, for $k\leq n$ any
character of $\Sym{n}$ is also a function on its subgroup $\Sym{k}$. Since the
inclusion $\Sym{k}\subseteq\Sym{n}$ is provided by augmenting any given
permutation by additional $n-k$ fixpoints, on the level of the cycle lengths
this corresponds to replacing any given partition $\pi\vdash k$ by $\pi\cup
1^{n-k}\vdash n$, just as it happens in~\eqref{eq:linear-characters}.
In the case of the spin groups the analogous sequence of inclusions
\[ \Spin{1}\subseteq \Spin{2} \subseteq \Spin{3} \subseteq \cdots \]
is naturally provided by considering first the infinite group generated by the
elements $z,t_1,t_2,\dots$ which fulfil the relations from~\cref{sec:spin} and
then viewing $\Spin{n}$ as its subgroup generated by $z,t_1,\dots,t_{n-1}$. It
turns out that also in this spin context the inclusion
$\Spin{k}\subseteq\Spin{n}$ is provided by replacing any given strict partition
$\pi\in\SP_{k}$ by $(\pi\cup 1^{n-k})\in\SP_{n}$, see~\cref{sec:conjugacy} for
more details.
It is more challenging to motivate quickly the specific choice of the
normalization constants in~\eqref{eq:linear-characters} and
\eqref{eq:projective-normalized}. In the case of the linear characters the
combinatorial factor $n^{\downarrow k}$ may be interpreted heuristically as the
number of ways the smaller symmetric group $\Sym{k}$ can be embedded in the
larger symmetric group $\Sym{n}$ or, more precisely, the number of ways any
$k$-element set can be embedded in an $n$-element set.
In the case of the spin characters a heuristic explanation is even more challenging; \emph{probably} it can be motivated by the work of Ivanov~\cite{Ivanov2004}.
The ultimate
criterion for verifying that the normalization constants in the definitions of
the normalized characters were selected \emph{in the right way} is postponed to
\cref{sec:KO-algebra,sec:KO-algebra-spin}. Roughly speaking, thanks to the right choice of the
normalization both $\Ch_\pi(\lambda_1,\lambda_2,\dots)$ and
$\ChSpin_{\pi}(\xi_1,\xi_2,\dots)$ can be identified with certain
\emph{polynomials} in the variables $\lambda_1,\lambda_2,\dots$ (respectively,
$\xi_1,\xi_2,\dots$).
\emph{Our goal in this paper is to find relationships between the family
$(\Ch_\class)$ of linear characters and its spin counterpart $(\ChSpin_\class)$. This
quite general goal can take various more specific incarnations which we will
discuss in the following.}
\subsection{The main result 1. Characters: linear vs spin}
Firstly, we might look for some explicit algebraic formulas which express the
linear characters $(\Ch_\class)$ in terms of the spin characters $(\ChSpin_\class)$
and vice versa.
The first step in this direction is to relate the set $\SP$ of strict partitions
to the set $\Part$ of partitions which are, respectively, the domain of the linear
characters $(\Ch_\class)$ and the domain of the spin characters $(\ChSpin_\class)$.
More specifically, for a strict partition $\xi\in\SP_n$ we consider its
\emph{double} $D(\xi)\in \Part_{2n}$. Graphically, $D(\xi)$ corresponds to a
Young diagram obtained by arranging the shifted Young diagram $\xi$ and its
`transpose' so that they nicely fit along the `diagonal', cf.~\cref{fig:double},
see also~\cite[Example~I.1.9(a)]{Macdonald1995}.
In~\cref{sec:spin-in-terms-of-linear} we will show several formulas which relate
linear and spin characters. The following is a particularly simple special case
which nevertheless gives a flavour of the general case.
\begin{theorem}[The main result for characters, special case]
\label{thm:main-special}
\ \\ For any odd integer $k\geq 1$ and any strict partition $\xi\in\SP$
\[ \Ch_{k}\big( D(\xi) \big) = 2\ \ChSpin_k(\xi). \]
\end{theorem}
We will prove this result (in wider generality) in~\cref{sec:proof}.
The results of this type reduce the problem of calculating the spin
characters to its linear counterpart for which often more tools are available
\cite{Rattan'Sniady2008,FeraySniady2011a,GouldenRattan2007,Biane2005/07,PetrulloSenato2011,DolegaFeraySniady2008}.
\subsection{The main result 2. Spin version of Stanley character formula}
\subsubsection{Colorings}
\label{sec:colorings}
Let $\sigma_1,\sigma_2\in\Sym{k}$ be permutations and
let $\lambda\in\Part$ be a Young diagram. Following~\cite{FeraySniady2011a}, we
say that $(f_1,f_2)$ is a \emph{coloring} of $(\sigma_1,\sigma_2)$ which is
compatible with $\lambda$ if the following two conditions are fulfilled:
\begin{itemize}
\item $f_i\colon C(\sigma_i) \to \Z_+$ is a function on the set of cycles of
$\sigma_i$ for each $i\in\{1,2\}$; we view the values of $f_1$ as columns of
$\lambda$ and the values of $f_2$ as rows;
\item whenever $c_1\in C(\sigma_1)$ and $c_2\in C(\sigma_2)$ are cycles which are not
disjoint ($c_1\cap c_2\neq \emptyset$), the box with Cartesian coordinates
$\big( f_1(c_1), f_2(c_2) \big)$ belongs to $\lambda$.
\end{itemize}
We denote by $N_{\sigma_1,\sigma_2}(\lambda)$ the number of such colorings of
$(\sigma_1,\sigma_2)$ which are compatible with $\lambda$.
\begin{figure}[htb]\centering
% \centerline{
\begin{tikzpicture}[scale=1]
\draw [->] (0,0) -- (5,0) node[anchor=west] {$x$};
\draw [->] (0,0) -- (0,3) node[anchor=west] {$y$};
\begin{scope}
\clip (0,0) -- (3,0) -- (3,1) -- (1,1) -- (1,2) -- (0,2);
\draw (0,0) grid (2,2);
\end{scope}
\draw[ultra thick] (0,0) -- (3,0) -- (3,1) -- (1,1) -- (1,2) -- (0,2) -- cycle;
\draw[line width=7pt, opacity=0.3, draw=red] (0.5,-0.5) node[anchor=north,opacity=1, text=black]{$f_1(V)$} -- (0.5,2.5);
\draw[line width=7pt, opacity=0.3, draw=blue] (2.5,-0.5) node[anchor=north,opacity=1, text=black]{$f_1(W)$} -- (2.5,1.5);
\draw[line width=7pt, opacity=0.3, draw=OliveGreen] (-0.5,0.5) node[anchor=east,opacity=1, text=black]{$f_2(\Pi)$} -- (3.5,0.5);
\draw[line width=7pt, opacity=0.3, draw=black] (-0.5,1.5) node[anchor=east,opacity=1, text=black]{$f_2(\Sigma)$} -- (1.5,1.5);
\end{tikzpicture}
% }
\caption{Graphical representation of the coloring~\eqref{eq:coloring} of the
permutations~\eqref{eq:example} which is compatible with the Young diagram
$\lambda=(3,1)$.} \label{fig:embed}
\end{figure}
\begin{exam}
Let
\begin{equation}
\label{eq:example}
\sigma_1=\underbrace{(1,5,4,2)}_{\textcolor{red}{V}}\underbrace{(3)}_{\textcolor{blue}{W}},\qquad \sigma_2 = \underbrace{(2,3,5)}_{\textcolor{OliveGreen}{\Pi}}\underbrace{(1,4)}_{\Sigma}.
\end{equation}
There are three pairs of cycles $(\sigma_1,\sigma_2)\in C(\sigma_1)\times
C(\sigma_2)$ with the property that $\sigma_1$ and $\sigma_2$ are not disjoint,
namely $(V,\Pi), (V,\Sigma), (W,\Pi)$.
It is now easy to check graphically (cf.~\cref{fig:embed}) that $(f_1,f_2)$ given by
\begin{equation}
\label{eq:coloring}
f_1(V)=1, \quad f_1(W)=3, \quad f_2(\Pi)=1, \quad f_2(\Sigma)=2
\end{equation}
is indeed an example of a coloring compatible with $\lambda=(3,1)$.
By considering four possible choices for the values of $f_2$ and counting the
choices for the values of $f_1$ one can verify that \[ N_{\sigma_1,\sigma_2}(\lambda)=
3^2 + 3 + 1 + 1 =14\qquad \text{for }\lambda=(3,1).\]
\end{exam}
\subsubsection{Linear Stanley character formula}
We will identify a given integer partition $\pi=(\pi_1,\dots,\pi_\ell)\vdash k$
with an arbitrary permutation $\pi\in\Sym{k}$ with the corresponding cycle
structure. For example, we may take
\[ \pi=(1,2,\dots,\pi_1)(\pi_1+1,\pi_1+2,\dots,\pi_1+\pi_2) \cdots\in\Sym{k}.\]
The following result turned out in the past to be a convenient tool for studying
asymptotic~\cite{FeraySniady2011a} and enumerative problems
\cite{DolegaFeraySniady2008} of the (linear) representation theory of the
symmetric groups.
\begin{theorem}[\cite{FeraySniady2011a}]
\label{thm:stanley-linear}
For any partition $\pi\in\Part_k$ and any Young diagram $\lambda\in\Part$
\begin{equation}
\label{eq:stanley-linear}
\Ch_\pi(\lambda) = \sum_{\substack{\sigma_1,\sigma_2\in\Sym{k} \\ \sigma_1 \sigma_2=\pi }}
(-1)^{\sigma_1} N_{\sigma_1,\sigma_2} (\lambda).
\end{equation}
\end{theorem}
\subsubsection{Spin Stanley character formula}
\looseness-1
For $\sigma_1,\sigma_2\in\Sym{k}$ we denote by \mbox{$|\sigma_1\vee \sigma_2|$}
the number of
orbits in the set $[k]=\{1,\dots,k\}$ under the action of the group $\langle
\sigma_1,\sigma_2\rangle$ generated by $\sigma_1,\sigma_2$. Equivalently, it is
the number of blocks of the set-partition $\sigma_1 \vee \sigma_2$ (which is the
minimal set-partition with respect to the refinement order which is greater than
each of the two set-partitions which correspond to the cycles of $\sigma_1$ and
$\sigma_2$).
One of the main results of the current paper is the following spin analogue of
\cref{thm:stanley-linear}.
\begin{theorem}[The main result: spin Stanley character formula]
\label{thm:spin-Stanley}
For any odd partition $\pi\in\OP_k$ and any strict partition $\xi\in\SP$
\begin{equation}
\label{eq:spin-stanley}
\ChSpin_\pi(\xi) = \sum_{\substack{\sigma_1,\sigma_2\in\Sym{k} \\ \sigma_1 \sigma_2=\pi }}
\frac{1}{2^{|\sigma_1\vee \sigma_2|}}\ (-1)^{\sigma_1}\ N_{\sigma_1,\sigma_2}\big( D(\xi) \big).
\end{equation}
\end{theorem}
The proof is postponed to~\cref{sec:proof-of-stanley}.
\subsubsection{Application: combinatorics of spin Kerov polynomials and Stanley polynomials}
In the context of the usual (linear) asymptotic representation theory of the
symmetric groups it is convenient to parametrize the set of Young diagrams by
\emph{free cumulants}~\cite{Biane1998}. Each linear character $\Ch_\pi$ can
then be expressed uniquely as a polynomial (called \emph{Kerov character
polynomial}) in these free cumulants. Kerov polynomials turned out to have a
very rich structure from the viewpoint of enumerative and algebraic
combinatorics
\cite{Biane2003,GouldenRattan2007,Biane2005/07,PetrulloSenato2011,DolegaFeraySniady2008,Sniady2013}.
Combinatorics of Kerov polynomials is intimately related to
\emph{multirectangular coordinates} of Young diagrams and the corresponding
\emph{Stanley character polynomials}, see~\cref{sec:multirectangular}.
Recently De Stavola~\cite{DeStavolaThesis} and the first-named author
\cite{Matsumoto2018} initiated investigation of spin counterparts of these
notions. In a %forthcoming publication~\cite{MatsumotoSniady2018c}
future work we will
present applications of~\cref{thm:spin-Stanley} to combinatorics of spin Kerov
polynomials and spin Stanley character polynomials.
\subsubsection{Application: bounds on spin characters}
The following character bound is a spin version of an analogous result for the
linear characters of the symmetric group~\cite{FeraySniady2011a}. It is a direct
application of~\cref{thm:spin-Stanley} and its proof follows the same line as
its linear counterpart~\cite{FeraySniady2011a}.
\begin{coro}
\label{cor:Annals}
There exists a universal constant $a>0$ with the property that for any integer
$n\geq 1$, any strict partition $\xi=(\xi_1,\xi_2 \dots)\in\SP_n$, and any odd partition
$\pi\in\OP_n$
\[
\left| \frac{\PHIeasy{\xi}{\pi}}{\PHIeasy{\xi}{1^n}} \right|
< \left[a \max\left( \frac{\xi_1}{n} , \frac{n-\ell(\pi)}{n} \right) \right]^{n-\ell(\pi)}. \]
\end{coro}
Several asymptotic results about (random) Young diagrams and tableaux which use
the inequality from~\cite{FeraySniady2011a} can be generalized in a rather
straightforward way to (random) \emph{shifted} Young diagrams and \emph{shifted}
tableaux thanks to~\cref{cor:Annals}. A good example is provided by the results
of Dousse and F{\'e}ray about the asymptotics of the number of skew standard Young
tableaux of prescribed shape~\cite{Dousse2017} which can be generalized in this
way to asymptotics of the number of \emph{shifted} skew standard Young tableaux.
\subsubsection{Open problem: towards irreducible representations of spin groups}
The proof of
the linear Stanley formula~\eqref{eq:stanley-linear} presented in
\cite{FeraySniady2011a} was found in the following way.
We attempted to
reverse-engineer the right-hand side of~\eqref{eq:stanley-linear} and to find
\begin{itemize}
\item some natural vector space $V$ with the basis indexed by combinatorial objects;
the space $V$ should be a representation of the symmetric group $\Sym{n}$ with
$n:=|\lambda|$, and
\item a projection $\Pi\colon V\to V$ such that $\Pi$ commutes
with the action of $\Sym{n}$ and such that it image $\Pi V$ is an irreducible
representation of $\Sym{n}$ which corresponds to the specified Young diagram
$\lambda$,
\end{itemize}
in such a way that the corresponding character of $\Pi V$ would coincide with
the right-hand side of~\eqref{eq:stanley-linear}.
Our attempt was successful: one could consider a vector space $V$ with the basis
indexed by fillings of the boxes of $\lambda$ with the numbers from $[n]$ with the
action of $\Sym{n}$ given by pointwise relabelling of the values in the boxes.
The projection $\Pi$ turned out to be the Young symmetrizer with the action
given by shuffling of the boxes in the rows and columns of $\lambda$. The
resulting representation $\Pi V$ clearly coincides with the Specht module, which
concluded the proof in~\cite{FeraySniady2011a}.
%\medskip
The structure of the right-hand side of~\eqref{eq:spin-stanley} (as well as the
right-hand side of~\eqref{eq:spin-stanley-nonoriented}) might be an indication
that an analogous reverse-engineering process could be applied to the spin case.
The result would be a very explicit construction of the irreducible spin
representations which would be an alternative to the somewhat complicated
approach of Nazarov~\cite{Nazarov1990}.
\subsection{Stanley formulas and enumeration of maps}
Some readers may have aesthetic objections against Equation
\eqref{eq:spin-stanley} related to the somewhat ugly factor
$\frac{1}{2^{|\sigma_1\vee \sigma_2|}}$ on the right-hand side. Our remedy to
this issue is~\cref{thm:maps} which avoids such factors. This result is just a
reformulation of~\cref{thm:spin-Stanley} in the language of \emph{non-oriented
maps which are orientable}. We start by introducing the notations which are
necessary to state this result.
\subsubsection{Maps} \label{sec:maps} We recall that a \emph{map}
\cite{Lando2004} is defined as a graph $\mathcal{G}$ drawn on a surface $\Sigma$
without boundary in such a way that each connected component of $\Sigma\setminus
\mathcal{G}$ (called \emph{face}) is homeomorphic to an open disc. In the
literature one often adds the requirement that the surface $\Sigma$ is
connected; we will not impose such a restriction.
Each map which we will consider is a \emph{bicolored map} which means that the
set of its vertices $\mathcal{V}=\mathcal{V}_\circ\sqcup \mathcal{V}_\bullet$ is
canonically decomposed into two classes (usually called white and black
vertices), with edges connecting only vertices of the opposite colors. Let
$F_1,\dots,F_\ell$ be the faces of the map; for the above reason of
bicoloration, each face $F_i$ is a polygon which consists of an even number
(say, $2\pi_i$) of edges, with neighbouring vertices painted in the alternating
colours. The integer partition $\pi=(\pi_1,\dots,\pi_\ell)$ is called the
\emph{face-type} of the map, see~\cref{fig:map-projective-plane}.
With this in mind, one can alternatively view a bicolored map as a collection
$\mathcal{P}$ of polygons with labelled edges and with the vertices painted in
the alternating colours (see~\cref{fig:polygons}), together with the information
which pairs of the edges should be glued together. Note that at first moment it
might appear that there are two ways to glue any given pair of edges (these two
ways differ by a twist a la M{\"o}bius), nevertheless the bicoloration of the
vertices makes the choice unique.
\subsubsection{Non-oriented maps}
\label{sec:sum-nonoriented}
The above discussion motivates the following definition. \emph{Summation over non-oriented maps
with a specified face-type $\pi$}~\cite[Section~3.4]{DolegaFeraySniady2014}
should be understood as follows: we start by fixing an appropriate collection $\mathcal{P}$ of
polygons with labelled edges, bicolored vertices, and with face-type $\pi$.
Then we consider all perfect matchings on the set of
all edges, and for each such a perfect matching we glue the corresponding pairs
of the edges. We sum over the resulting map.
\newcommand{\faceA}{red!50}
\newcommand{\faceB}{blue}
\newcommand{\faceAfill}{red!10}
\newcommand{\faceBfill}{blue!20}
\begin{figure}[htb]
\begin{tikzpicture}[
yscale=-0.5,xscale=0.5,
black/.style={circle,thick,draw=black,fill=white,inner sep=4pt},
white/.style={circle,draw=black,fill=black,inner sep=4pt},
connection/.style={draw=black,black,auto}
]
\small
\fill[pattern color=\faceAfill,pattern=north west lines] (0*36:5) -- (1*36:5) -- (2*36:5) -- (3*36:5) -- (4*36:5) -- (5*36:5) -- (6*36:5) -- (7*36:5) -- (8*36:5) -- (9*36:5);
\draw (0*36:5) node (b1) [black] {};
\draw (1*36:5) node (b2) [white] {};
\draw (2*36:5) node (b3) [black] {};
\draw (3*36:5) node (b4) [white] {};
\draw (4*36:5) node (b5) [black] {};
\draw (5*36:5) node (b6) [white] {};
\draw (6*36:5) node (b7) [black] {};
\draw (7*36:5) node (b8) [white] {};
\draw (8*36:5) node (b9) [black] {};
\draw (9*36:5) node (b10) [white] {};
\draw (0.5*36:4.1) node {$1$};
\draw (1.5*36:4.1) node {$2$};
\draw (2.5*36:4.1) node {$3$};
\draw (3.5*36:4.1) node {$4$};
\draw (4.5*36:4.1) node {$5$};
\draw (5.5*36:4.1) node {$6$};
\draw (6.5*36:4.1) node {$7$};
\draw (7.5*36:4.1) node {$8$};
\draw (8.5*36:4.1) node {$9$};
\draw (9.5*36:4.1) node {$10$};
\draw[connection,decoration={
markings,
mark=at position 0.5 with {\arrow[red,line width=1.5mm]{right to}}},
postaction={decorate}]
(b1) to (b2);
\draw[red, line width=0.5mm,->] (b1)+(-2,0) to (b1);
\draw[connection] (b2) to (b3);
\draw[connection] (b3) to (b4);
\draw[connection] (b4) to (b5);
\draw[connection] (b5) to (b6);
\draw[connection] (b6) to (b7);
\draw[connection] (b7) to (b8);
\draw[connection] (b8) to (b9);
\draw[connection] (b9) to (b10);
\draw[connection] (b10)to (b1);
\begin{scope}[shift={(10,0)},scale=0.5]
\fill[\faceBfill] (0:5) -- (90:5) -- (180:5) -- (270:5);
\draw (90*0:5) node (b1) [black] {};
\draw (90*1:5) node (b2) [white] {};
\draw (90*2:5) node (b3) [black] {};
\draw (90*3:5) node (b4) [white] {};
\draw (90*0.5:2.2) node {$A$};
\draw (90*1.5:2.2) node {$B$};
\draw (90*2.5:2.2) node {$C$};
\draw (90*3.5:2.2) node {$D$};
\draw[connection,decoration={
markings,
mark=at position 0.5 with {\arrow[blue,line width=1.5mm]{right to}}},
postaction={decorate}]
(b1) to (b2);
\draw[blue, line width=0.5mm,->] (b1)+(-3,0) to (b1);
\draw[connection] (b2) to (b3);
\draw[connection] (b3) to (b4);
\draw[connection] (b4) to (b1);
\end{scope}
\end{tikzpicture}
\caption{Example of a collection of polygons with face-type $(5,2)$. Arrows
(pointing at white vertices between the edges $1$ and $10$, as well as between
$A$ and $D$) indicate the roots; harpoons (at the edges $1$ and $A$)
indicate the directions of rotation in each
polygon, see~\cref{sec:non-oriented-revisited}.} \label{fig:polygons}
\vspace{3ex}
\begin{tikzpicture}[
yscale=-0.6,xscale=0.6,
black/.style={circle,thick,draw=black,fill=white,inner sep=4pt},
white/.style={circle,draw=black,fill=black,inner sep=4pt},
connection/.style={draw=black,black,auto}
]
\scriptsize
\begin{scope}
\clip (0,0) rectangle (10,10);
\fill[pattern color=\faceAfill,pattern=north west lines] (0,0) rectangle (10,10);
\fill[\faceBfill] (5,5) rectangle (8,8);
\draw (2,7) node (b1) [black] {};
\draw (12,3) node (b1prim) [black] {};
\draw (3,2) node (w1) [white] {};
\draw (-7,8) node (w1prim) [white] {};
\draw (5,5) node (AA) [black] {};
\draw (8,5) node (BA) [white] {};
\draw (5,8) node (AB) [white] {};
\draw (8,8) node (BB) [black] {};
\draw[connection] (w1) to node [swap,pos=0.6] {$4$} node [pos=0.6] {$9$} (AA);
\draw[connection] (AA) to node [swap] {$5$} node {$D$} (AB);
\draw[connection] (AB) to node [swap] {$6$} node {$C$} (BB);
\draw[connection] (BB) to node [swap] {$7$} node {$B$} (BA);
\draw[connection,decoration={
markings,
mark=at position 0.5 with {\arrow[blue,line width=1.5mm]{right to}}},
postaction={decorate}] (AA) to node {$8$} node [swap] {$A$} (BA);
\draw[blue, line width=0.5mm,->] (AA)+(1,1) to (AA);
\draw[connection] (w1) to node [swap,pos=0.5] {$10$} node [pos=0.425] {$2$} (b1prim);
\draw[connection] (w1prim) to node [swap,pos=0.87] {$2$} node [pos=0.95] {$10$} (b1);
\draw[connection,decoration={
markings,
mark=at position 0.5 with {\arrow[red,line width=1.5mm]{left to}}},
postaction={decorate}] (b1) to node [pos=0.575] {$1$} node [swap,pos=0.5] {$3$} (w1);
\draw[red, line width=0.5mm,->] (b1)+(-1,-1) to (b1);
\end{scope}
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>}}},
postaction={decorate}]
(0,0) -- (10,0);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>}}},
postaction={decorate}]
(10,10) -- (0,10);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>>}}},
postaction={decorate}]
(10,0) -- (10,10);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>>}}},
postaction={decorate}]
(0,10) -- (0,0);
\end{tikzpicture}
\caption{Example of a \emph{non-oriented map} with face-type $(5,2)$ drawn on the projective plane.
The left side of the square should be glued with a twist to the right side,
as well as bottom to top (also with a twist), as indicated by arrows.
This map has been obtained by gluing the edges of the polygons from Figure~\ref{fig:polygons}. }
\label{fig:map-projective-plane}
\end{figure}
\subsubsection{Colorings revisited}
The notion of \emph{number of colorings} $N_{\sigma_1,\sigma_2}(\lambda)$ was
introduced in~\cref{sec:colorings} in the context of a pair of permutations. We
review how to generalize it to maps and, more generally, bicolored graphs.
Let $\mathcal{G}$ be a bicolored graph with the vertex set $\mathcal{V}=\mathcal{V}_\circ\sqcup \mathcal{V}_\bullet$ and
let $\lambda\in\Part$ be a Young diagram.
We say that $(f_1,f_2)$ is a \emph{coloring} of $\mathcal{G}$ which is
compatible with $\lambda$ if the following two conditions are fulfilled:
\begin{itemize}
\item $f_1\colon \mathcal{V}_\circ \to \Z_+$ is a function on the set of white
vertices while $f_2\colon \mathcal{V}_\bullet \to \Z_+$ is a function on the
set of black vertices;
\item whenever vertices $v_1\in \mathcal{V}_\circ$ and $v_2\in
\mathcal{V}_\bullet$ are connected by an edge, the box with Cartesian
coordinates $\big( f_1(v_1), f_2(v_2) \big)$ belongs to $\lambda$.
\end{itemize}
We denote by $N_{\mathcal{G}}(\lambda)$ the number of such colorings of
$\mathcal{G}$ which are compatible with $\lambda$.
\begin{rema}
In~\cref{sec:labels} we shall explain how to associate a map $\mathcal{G}$ to a
pair $(\sigma_1,\sigma_2)$ of permutations; the above definition of
$N_{\mathcal{G}}(\lambda)$ is compatible with this correspondence in the sense
that
\[ N_{\sigma_1,\sigma_2}(\lambda)= N_{\mathcal{G}}(\lambda),\]
where the left-hand side should be understood in the sense of~\cref{sec:labels}.
\end{rema}
\subsubsection{Spin Stanley formula and non-oriented, orientable maps}
The following is a reformulation of~\cref{thm:spin-Stanley}.
\begin{theorem}
\label{thm:maps}
For any odd partition $\pi\in\OP_k$ and any strict partition $\xi\in\SP$
\begin{equation}
\label{eq:spin-stanley-nonoriented} 2^{\ell(\pi)}\ \ChSpin_\pi (\xi)
= \sum_M (-1)^{|\pi|- |\mathcal{V}_\circ(M)|}\ N_M\big( D(\xi) \big)
\end{equation}
where
the sum runs over non-oriented, orientable bicolored maps with the face-type $\pi$, as in~\cref{sec:sum-nonoriented}.
\end{theorem}
The proof is postponed to~\cref{sec:proof-of-maps}.
\begin{table}[htb]\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|>{\PBS\raggedright}p{4.1cm}|>{\PBS\raggedright}p{3.6cm}|>{\PBS\raggedright}p{3.8cm}|}
\hline
\textsc{type of characters} & \textsc{type of maps} & \textsc{references}
\\ \hline \hline
Linear characters\break of the symmetric groups & Oriented maps &
\cite[Sections~6.4--6.5]{Sniady2013}, \cite[Section~8]{Sniady2016}
\\
\hline
Spin characters\break of the symmetric groups & Non-oriented maps\break which are orientable & Current paper,\break~\cref{thm:maps}
\\ \hline
Zonal characters & \multirow{2}{3.5cm}{Non-oriented maps} & \multirow{2}{*}{\cite[Section~5]{Feray2011}} \\ \cline{1-1}
Symplectic zonal characters & & \\ \hline
\multirow{2}{3.6cm}{Top-degree \\ of Jack characters} & Oriented,\break connected maps\break with arbitrary\break face structure & \cite[Sections~1.4.2--1.4.3]{SniadyTopDegree}
\\
\cline{2-3}
& Non-oriented maps\break with prescribed\break face structure& \cite[Section~4]{Czyzewska-Jankowska2017} \\ \hline
Jack characters in the generic case &
conjecturally:\break non-oriented maps\break
(counted with some unknown weight) & partial results: \cite{DolegaFeraySniady2014}
\\ \hline
\end{tabular}
\vspace{2ex}
\caption{Classes of functions on the set of (shifted) Young diagrams for which
some version of Stanley character formula is known
(or conjectured in the case of Jack characters) and the corresponding class
of maps over which the summation is performed.}
\label{tab:table}
\end{table}
\subsubsection{Stanley formulas and maps}
\cref{tab:table} summarizes some known and some hypothetical Stanley formulas.
It seems that there is a correspondence between some natural functions on the
set of (shifted) Young diagrams and natural classes of bicolored maps. Is there
some general pattern? Are there some natural classes of maps which are missing
in this table?
\subsection{The main result 3. Kerov--Olshanski algebra: linear vs spin}
\subsubsection{Kerov--Olshanski algebra}
\label{sec:KO-algebra}
The usual (linear) Kerov--Olshanski algebra $\KO$
\cite{Kerov1994,Hora2016} (also known under the less compact name \emph{algebra
of polynomial functions on the set of Young diagrams}; in the monograph
\cite[Section~7]{Meliot2017} it is referred to as \emph{Ivavov--Kerov algebra})
is an important tool in the (linear) asymptotic representation theory of the
symmetric groups. One of its advantages comes from the fact that it can be
characterized in several equivalent ways (for example as the algebra $\Lambda^*$
of \emph{shifted symmetric functions}); it also has several convenient linear
and algebraic bases which are related to various viewpoints and aspects of the
asymptotic representation theory.
For the purposes of the current paper, Kerov--Olshanski algebra
\[ \KO := \spanrm\{ \Ch_\class : \class \in \Part \} \]
may be defined as
the linear span of the normalized linear characters of the symmetric groups.
\subsubsection{Spin Kerov--Olshanski algebra}
\label{sec:KO-algebra-spin}
We define the \emph{spin} Kerov--Olshanski algebra
(maybe \emph{Ivanov algebra} would be an even better name)
\begin{equation}
\label{eq:Gamma}
\Gamma := \spanrm\{ \ChSpin_\class : \class \in \SP \}
\end{equation}
as the linear span of spin characters~\cite[Section~6]{Ivanov2004}.
Ivanov proved
that the elements of $\Gamma$ can be identified with
\emph{supersymmetric polynomials}, thus $\Gamma$ is a unital, commutative algebra.
\subsubsection{Double of a function. Kerov--Olshanski algebra: linear vs spin}
If $F\colon \Part\to \Q$ is a function on the
set of partitions, we define its \emph{double} as the function $\double F \colon \SP \to \Q$ on the
set of \emph{strict} partitions
\[ \left( \double F \right)(\xi) := F\big( D(\xi) \big) \qquad \text{for }\xi\in\SP\]
given by doubling of the argument.
%\medskip
The following simple result reduces some questions about the spin
Kerov--Olshanski algebra $\Gamma$ to its linear counterpart $\KO$.
\begin{theorem}
\label{prop:isomorphism}
The map
\begin{equation}
\label{eq:isomorphism}
\double \colon \KO/\ker \double \to \Gamma
\end{equation}
is an isomorphism of algebras.
\end{theorem}
The proof is postponed to~\cref{sec:proof-of-isomorphism}.
\subsubsection{Filtrations: linear vs spin}
\label{sec:filtrations}
In applications it is often convenient to equip Kerov--Olshanski algebra with
this or another filtration which is tailored for the specific asymptotic regime
one is interested in; several such filtrations were considered in the
literature. In order to stay focused we will concentrate on a specific choice of
such a filtration $\F_0\subseteq \F_1 \subseteq \cdots \subseteq \KO$ given by
\begin{equation}
\label{eq:filtration}
\F_k := \spanrm\{ \Ch_\class : \class \in \Part,\quad |\class|+\ell(\class)\leq k \},
\end{equation}
\looseness-1
where $\ell(\class)=\ell$ denotes the number of parts of
$\class=(\class_1,\dots,\class_\ell)$.
This specific choice is motivated by investigation of asymptotics of (random)
Young diagrams and tableaux in the scaling in which they grow to infinity in
such a way that they remain \emph{balanced}~\cite{Biane1998,Sniady2006}.
%\medskip
We define its spin counterpart as the family of vector spaces
$\Gcal_0\subseteq \Gcal_1 \subseteq \cdots \subseteq \Gamma$ defined by
\begin{equation}
\label{eq:filtration2}
\Gcal_k := \spanrm\{ \ChSpin_\class : \class \in \OP,\quad |\class|+\ell(\class)\leq k \}.
\end{equation}
Note that $|\class|+\ell(\class)$ is always an even integer for any $\class \in
\OP$; it follows therefore that $\Gcal_{2k+1}=\Gcal_{2k}$ holds for any integer $k\geq 0$.
Informally speaking, this means that only the \emph{`even'} part of the family
$\Gcal_0\subseteq \Gcal_1 \subseteq \cdots \subseteq \Gamma$ contains interesting
information.
%\medskip
The following result provides a direct link between the families $(\F_i)$ and
$(\Gcal_i)$ via the isomorphism~\eqref{eq:isomorphism}. Also its proof (postponed
to~\cref{sec:proof-of-filtration}) makes use of this isomorphism.
\begin{theorem}
\label{thm:filtration}
The family $(\Gcal_k)$ is a filtration on the algebra $\Gamma$.
Furthermore, $\Gcal_k = \double(\F_k)$ for any integer $k\geq 0$.
\end{theorem}
\subsubsection{Application. Gaussian fluctuations of partitions: linear vs spin}
Our motivation for studying the filtration $(\Gcal_i)$ comes from investigation of
random strict partitions related to projective asymptotic representation theory
of the symmetric groups.~\cref{thm:filtration} plays a prominent role in our
forthcoming paper~\cite{Sniady2019} which is devoted to this topic.
A convenient tool for proving Gaussianity of fluctuations of random partitions
is \emph{approximate factorization property for characters}
\cite{Sniady2006,DolegaSniady2018} which is formulated in the language of
certain \emph{cumulants}. In the case of the \emph{linear} characters of the
symmetric groups this property was known to be true~\cite{Sniady2006}.
In the aforementioned forthcoming paper~\cite{Sniady2019} we will show how
to reformulate the results of the current paper in a more abstract way which
allows to relate the cumulants for \emph{linear} characters to their \emph{spin}
counterparts. In this way we will show how the approximate factorization
property of \emph{linear} characters of the symmetric groups implies directly
its \emph{spin} counterpart.
\subsection{Structure of the paper}
\cref{sec:spin-characters} summarizes some fine issues related to the definition
of the spin characters.
In~\cref{sec:spin-in-terms-of-linear} we prove our starting tool,
\cref{thm:general_characters}, which expresses the linear characters $(\Ch_\pi)$
in terms of the spin characters $(\ChSpin_\pi)$. As a consequence, we also prove
\cref{theo:spin-in-linear} which gives roughly the opposite: the spin characters
$(\ChSpin_\pi)$ in terms of the linear characters $(\Ch_\pi)$.
\cref{sec:proof-of-stanley} is devoted to the proof of~\cref{thm:spin-Stanley},
\ie spin Stanley character formula.~\cref{sec:proof-of-maps} is devoted to the
proof of~\cref{thm:maps}, \ie spin Stanley formula in terms of non-oriented,
orientable maps.
\looseness-1
In~\cref{sec:multirectangular} we recall the shifted version of
\emph{multirectangular coordinates} and in~\cref{sec:asymptotic-stanley} we
apply these coordinates for asymptotics of spin characters. In
\cref{sec:proof-of-isomorphism} we apply these results in order to prove
\cref{prop:isomorphism,thm:filtration} which provide the link between (the
filtration on) the linear Kerov--Olshanski algebra and its spin counterpart.
%\medskip
This article is the full version of a $12$-page \emph{extended abstract}
\cite{MatsumotoSniadyFPSAC} which will be published in the proceedings of the conference
\emph{Formal Power Series and Algebraic Combinatorics 2019}.
\section{Spin characters, spin representations}
\label{sec:spin-characters}
In order to keep the Introduction lightweight we decided to postpone the
discussion of some subtle technical issues related to the definition of spin
characters and spin representations until the current section.
Our presentation is based on
\cite{Stembridge1989,Wan2012,Kleshchev2005,Ivanov2004}.
\subsection{Conjugacy classes of \texorpdfstring{$\widetilde{\mathfrak{S}}_{n}$}{Sn}}
We denote by $\SP_n^+$ (respectively, $\SP_n^-$) the set of
strict partitions $\xi\in\SP_n$ with the property that
$n-\ell(\xi)$ is even (respectively, odd).
%\medskip
For a partition $\pi\vdash n$ we denote by $C_\pi\subset \Spin{n}$ the set of
elements of the spin group which are mapped --- under the canonical homomorphism
$\Spin{n}\to\Sym{n}$ --- to permutations with the cycle-type given by $\pi$.
%\medskip
Schur~\cite{Schur1911}
proved the following dichotomy for $\pi\vdash n$:
\begin{itemize}
\item if one of the following two conditions is fulfilled:
\begin{itemize}%[label=\ding{212}]
\item $\pi\in \OP_n$, or
\item $\pi\in \SP_n^-$
\end{itemize}
then $C_\pi$ splits into a pair of conjugacy classes of $\Spin{n}$ which will
be denoted by $C_\pi^\pm$;
\item otherwise, $C_\pi$ is a conjugacy class of $\Spin{n}$.
\end{itemize}
\subsection{Conjugacy classes and spin characters}
\label{sec:conjugacy}
Any spin character vanishes on the conjugacy class $C_\pi$ which does not split,
cf.~\cite[p.~95]{Stembridge1989}. For this reason, \emph{from the viewpoint of
the spin character theory only the conjugacy classes $C_\pi^\pm$ are
interesting}.
%\medskip
Spin representations are exactly the ones which map the central element
$z\in\Spin{n}$ to $-\operatorname{Id}\in\GL(V)$. Since $C_\pi^- = z C_\pi^+$, it
follows that the value of any spin character on $C_\pi^-$ is the opposite of its
value on $C_\pi^+$. For this reason, \emph{from the viewpoint of the spin
character theory the conjugacy classes $C_\pi^-$ are redundant and it is enough
to consider the character values only on the conjugacy classes $C_\pi^+$}.
%\medskip
From the viewpoint of the asymptotic representation theory it is natural to
consider some \emph{sequence} of groups together with some natural inclusions;
in our case this is the sequence
\[ \Spin{1} \subset \Spin{2} \subset \Spin{3} \subset \cdots \]
of spin groups. Such a setup allows to relate a conjugacy class of a smaller
group to some conjugacy class in the bigger group and, in this way, to evaluate
the irreducible characters of the bigger group on the conjugacy classes of the
smaller one.
Regretfully, the conjugacy classes $C_\pi^+$ which correspond to $\pi\in\SP_n^-$
do not behave nicely under such inclusions. Indeed, on the level of the
symmetric groups the inclusion $\Sym{n}\subset \Sym{n+k}$ corresponds to adding
$k$ fixpoints to a given permutation; in other words the set $C_\pi\subset
\Spin{n}$ corresponds to $C_{\pi,1^k}\subset \Spin{n+k}$ and the latter does not
split because $(\pi,1^k)\notin \SP_n$ (at least for $k\geq 2$) and
$(\pi,1^k)\notin \OP_n$ (because $n-\ell(\pi)$ is odd which implies that at
least one part of $\pi$ is even).
For this reason, \emph{for the purposes of the asymptotic representation theory
it is enough to consider only the conjugacy classes $C_\pi^+$ for
$\pi\in\OP_n$.}
\subsection{Irreducible spin representations}
\label{sec:irreducible-spin-representations}
The relationship between strict partitions and the irreducible spin
representations of the symmetric groups is \emph{not} a bijective one.
Nevertheless, as we shall discuss below, this non-bijectivity can be ignored to
large extent.
%\medskip
More specifically (see~\cite[p.~235]{Schur1911} and~\cite[Theorem~7.1]{Stembridge1989}), each $\xi\in\SP_n^+$ corresponds to a \emph{single}
irreducible representation.
We denote by its character by $\phi^{\xi}$.
%\medskip
On the other hand, each $\xi\in\SP_n^-$ corresponds to a \emph{pair} of
irreducible spin representations; we denote their characters by $\phi^{\xi}_+$
and $\phi^{\xi}_-$. These two characters coincide on the conjugacy classes
$C^{\pm}_{\pi}$ over $\pi\in\OP_n$. For the purposes of the current paper we do not
need to evaluate the characters on $C^{\pm}_{\pi}$ for $\pi\in \SP_n^-$; for
this reason we do not have to distinguish between them and we may denote them by
the same symbol $\phi^\xi$.
\subsection{Spin characters: conclusion}
\label{appendix:conclusion}
For all partitions $\xi\in\SP_n$, $\pi\in\OP_n$ the
value of the projective character
\[ \phi^\xi(\pi)= \Tr \irrepSp^\xi(c^\pi) \]
is well defined, where $c^\pi\in C_\pi^+$ is a representative of the of the
conjugacy class $C_\pi^+$, cf.~\cite[Eq.~(2.1)]{Stembridge1989}.
\subsection{Spin characters and symmetric functions}
In the context of the \emph{theory of symmetric functions} it is more
convenient to pass from spin characters $\phi^\xi$ to some other quantities which
are denoted by $X^\lambda$. We shall review them in the following.
%\medskip
Following~\cite[Section~III.7]{Macdonald1995} we denote by $X(t)$ the transition
matrix between the basis of Newton's power-sum symmetric functions $(p_\pi)$ and
the basis of Hall--Littlewood functions:
\begin{equation}
\label{eq:transition-matrix-X}
p_\pi = \sum_{\lambda} X^\lambda(\pi; t) P_\lambda(t).
\end{equation}
For some special choices of the parameter $t$ the quantities $X(t)$ turn out to
have interesting interpretations. For example, in the case $t=0$ we recover the
linear characters of the symmetric groups $\chi^\lambda(\pi)=X^\lambda(\pi;0)$
with $\lambda,\pi \in \Part_n$, cf.~\cite[III.(7.2)]{Macdonald1995}.
From the following on we shall concentrate on another special case \mbox{$t=-1$} for
which Hall--Littlewood functions turn out to be related to Schur's $P$- and
$Q$-functions~\cite[Section~III.8]{Macdonald1995}. In this specific setup the
defining property~\eqref{eq:transition-matrix-X} of the transition matrix
$X=X(-1)$ takes the form
\begin{align}
\nonumber
p_\pi &= \sum_{\xi \in \SP_n} X^\xi(\pi) P_\xi \\
\intertext{for $n=|\pi|$, or equivalently, for $n=|\xi|$}
\label{eq:Frobenius-spin}
Q_\xi &= \sum_{\nu \in \OP_n}
2^{\ell(\nu)} z_\nu^{-1}\ \XX{\xi}{\nu}\ p_\nu,
\end{align}
where
$P_\xi,Q_\xi$ are the Schur's $P$- and $Q$-functions,
as in~\cite[Example~III.8.11(c)]{Macdonald1995}
and
\[ z_\class = \prod_{j\geq 1} j^{m_j(\class)} m_j(\class)!, \]
where $m_j(\class)$ denotes the \emph{multiplicity} of $j$ in the partition
$\class$.
%\bigskip
Following Ivanov~\cite[Section~2]{Ivanov2004}, given $\xi\in\SP_n$ we define a function on $\OP_n$
\[ \widetilde{\phi}^\xi=
\begin{cases}
\phi^\xi & \text{if } \xi \in \SP_n^+, \\
\frac{\phi^\xi_+ + \phi^\xi_-}{\sqrt{2}}=
\sqrt{2}\ \phi^\xi & \text{if } \xi \in \SP_n^-.
\end{cases}
\]
Then the link between such spin characters $\widetilde{\phi}$ and the transition
matrix $X$ is provided by the equality
\[\XX{\xi}{\pi}=2^{\frac{\ell(\xi)-\ell(\pi)}{2}}\ \PHI{\xi}{\pi},\]
cf.~\cite[Proposition~3.3]{Ivanov2004}.
With these notations
\eqref{eq:projective-normalized} can be rewritten as
\[
\ChSpin_\class(\xi):=\begin{cases}
n^{\downarrow k}\
\ \frac{ \XX{\xi}{\class\cup 1^{n-k}}}{\XX{\xi}{1^{n}}} =
n^{\downarrow k}\ 2^{\frac{k-\ell(\class)}{2}}
\ \frac{ \PHI{\xi}{\class\cup 1^{n-k}}}{ \PHI{\xi}{1^{n}}}
& \text{if } n\geq k, \\
0 & \text{otherwise.}
\end{cases}
\]
\section{Spin characters in terms of linear characters}
\label{sec:spin-in-terms-of-linear}
\subsection{The main result 1}
We were not able to find this theorem in the literature and we believe it is
new. Note, however, that this result is essentially only a reformulation of the
classical equality~\eqref{eq:Schur_SchurQ} which gives a relation between Schur
polynomial $s_{D(\xi)}$ and Schur's $Q$-function $Q_\xi$.
We feel that this result is rather unexpected because the spin characters are
precisely those that are \emph{not} obtained by lifting the usual linear
characters of the symmetric group; for this reason one should not expect any
easy links between the linear and the spin characters.
\begin{theorem}[The main result 1: linear in terms of spin]
\label{thm:general_characters}
For any $\class\in\OP$ and $\xi\in\SP$,
\begin{equation}
\label{eq:spin-vs-linear}
\Ch_\class\big( D(\xi)\big) =
\sum_{\sett \subseteq [\ell(\class)]}
\ChSpin_{\class(\sett)}(\xi)\ \ChSpin_{\class(\sett^c)}(\xi)
\end{equation}
where $\class(\sett)=(\class_{i_1}, \class_{i_2},\dots, \class_{i_r})$ for
$\sett=\{i_1] (b1)+(-2,0) to (b1);
\draw (1*360/14:7) node (b2) [white] {};
\draw (2*360/14:7) node (b3) [black] {};
\draw (3*360/14:7) node (b4) [white] {};
\draw (4*360/14:7) node (b5) [black] {};
\draw (5*360/14:7) node (b6) [white] {};
\draw (6*360/14:7) node (b7) [black] {};
\draw (7*360/14:7) node (b8) [white] {};
\draw (8*360/14:7) node (b9) [black] {};
\draw (9*360/14:7) node (b10) [white] {};
\draw (10*360/14:7) node (b11) [black] {};
\draw (11*360/14:7) node (b12) [white] {};
\draw (12*360/14:7) node (b13) [black] {};
\draw (13*360/14:7) node (b14) [white] {};
\draw (-0.5*360/14:6) node {$1$};
\draw (-2.5*360/14:6) node {$2$};
\draw (-4.5*360/14:6) node {$3$};
\draw (-6.5*360/14:6) node {$4$};
\draw (-8.5*360/14:6) node {$5$};
\draw (-10.5*360/14:6) node {$6$};
\draw (-12.5*360/14:6) node {$7$};
\draw[connection] (b1) to (b2);
\draw[connection] (b2) to (b3);
\draw[connection] (b3) to (b4);
\draw[connection] (b4) to (b5);
\draw[connection] (b5) to (b6);
\draw[connection] (b6) to (b7);
\draw[connection] (b7) to (b8);
\draw[connection] (b8) to (b9);
\draw[connection] (b9) to (b10);
\draw[connection] (b10)to (b11);
\draw[connection] (b11) to (b12);
\draw[connection] (b12) to (b13);
\draw[connection] (b13) to (b14);
\draw[connection] (b14) to (b1);
\draw [ultra thick,<-] (-3,-2) arc(-160:160:1);
\end{tikzpicture}
\caption{A collection of oriented polygons of face type $(7)$. The labeling of
the edges corresponds to the permutation $\pi=(1,2,3,4,5,6,7)\in\Sym{7}$. The
root is indicated by the thick blue arrow. The orientation is indicated by the
circular arrow.} \label{fig:polygon-oriented}
\vspace{3ex}
\centering
\begin{tikzpicture}[scale=0.6,
white/.style={circle,draw=black,fill=white,inner sep=4pt},
black/.style={circle,draw=black,fill=black,inner sep=4pt},
connection/.style={draw=black!80,black!80,auto}
]
\footnotesize
\begin{scope}
\clip (0,0) rectangle (10,10);
\fill[blue!3] (0,0) rectangle (10,10);
\draw [ultra thick,<-] (4,8) arc(-160:160:1);
\draw (3,5) node (b1) [black] {};
\draw (b1) +(10,0) node (b1prim) [black] {};
\draw (b1) +(1,-3) node (b1-se) [white] {};
\draw (b1) +(-1,-3) node (b1-sw) [white] {};
\draw (8,8) node (b2) [black] {};
\draw (b2) +(0,-10) node (b2prim) [black] {};
\draw (b2) +(-10,0) node (b2prim2) [black] {};
\draw (6,5) node (w1) [white] {};
\draw (w1) +(0,10) node (w1prim) {};
\draw (12,7) node (w2) [white] {};
\draw (w2) +(-10,0) node (w2prim) [white] {};
\draw[connection,pos=0.2] (b2) to node {\textcolor{black}{$4$}} node [swap] {} (w1prim);
\draw[connection,pos=0.666] (b2prim) to node {\textcolor{black}{$4$}} node [swap] {} (w1);
\draw[connection,pos=0.2] (b2) to node {\textcolor{black}{$6$}} node [swap] {} (w2);
\draw[connection,pos=0.7] (b2prim2) to node {\textcolor{black}{$6$}} node [swap] {} (w2prim);
\draw[connection] (b1) to node {\textcolor{black}{$2$}} node [swap] {} (b1-sw);
\draw[connection] (b1) to node {\textcolor{black}{$3$}} node [swap] {} (b1-se);
\draw[connection] (b1) to node {\textcolor{black}{$5$}} node [swap] {} (w1);
\draw[connection] (w1) to node {} node [swap] {\textcolor{black}{$7$}} (b2);
\draw[connection] (w2) to node {} node [swap] {\textcolor{black}{$1$}} (b1prim);
\draw[connection] (w2prim) to node {} node [swap] {\textcolor{black}{$1$}} (b1);
\draw[blue, line width=1mm,->] (w2prim)+(-1,-1) to (w2prim);
\end{scope}
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>}}},
postaction={decorate}]
(0,0) -- (10,0);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>}}},
postaction={decorate}]
(0,10) -- (10,10);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>>}}},
postaction={decorate}]
(0,0) -- (0,10);
\draw[very thick,decoration={
markings,
mark=at position 0.666 with {\arrow{>>}}},
postaction={decorate}]
(10,0) -- (10,10);
\end{tikzpicture}
\caption{Example of an \emph{oriented map}. The map is drawn on a torus: the
left side of the square should be glued to the right side, as well as bottom to
top, as indicated by the arrows. The thick blue arrow indicates the root of the
face. The orientation is indicated by the circular arrow. This map was created
by glueing the edges of the oriented collection of polygons from
\cref{fig:polygon-oriented}.} \label{fig:map-kerov}
\end{figure}
\subsection{Oriented maps}
\label{sec:oriented}
We say that a map is \emph{oriented} if the surface $\Sigma$ is orientable \emph{and}
on each connected component some orientation is selected, see~\cref{fig:map-kerov}.
\emph{Summation over oriented maps with a specified face-type $\pi$}
should be understood as follows: we start by fixing an
appropriate collection $\mathcal{P}$ of polygons with labelled edges, bicolored
vertices, and with face-type $\pi$. \emph{Additionally, on each polygon we
choose some orientation (``which direction of rotation should be understood as
clockwise''),} see~\cref{fig:polygon-oriented}. Then we consider \emph{only}
these perfect matchings on the set of all edges, which result with a glueing of
the polygons with the property that the originally selected orientations on the
polygons are consistent when one crosses an edge. We sum over the resulting
oriented map.
\subsection{Labels on oriented maps}
\label{sec:labels}
In the context of the \emph{oriented} maps it is more convenient to label only
some edges of the oriented polygons from~$\mathcal{P}$; more specifcially we
label only these edges which --- if one traverses the edges in the clockwise
cyclic order --- start with a white vertex, see~\cref{fig:polygon-oriented}.
The labelling can be encoded by a permutation $\pi$, the cycles of which correspond
to the cyclic order of the labels around the polygons (in the clockwise direction),
see~\cref{fig:polygon-oriented}.
With this convention, the orientations of the polygons are consistent in the
map obtained by glueing the edges of $\mathcal{P}$ if and only if in each pair
of glued edges exactly one carries a label, see~\cref{fig:map-kerov}.
%\medskip
The structure of such an oriented map can be uniquely recovered from the pair of
permutations $\sigma_1, \sigma_2$, where the cycles of $\sigma_1$ encode the
counterclockwise cyclic order around the white vertices while the cycles of
$\sigma_2$ encode the counterclockwise cyclic order around the black vertices.
For example, for the map from~\cref{fig:map-kerov} we have
\begin{align*}
\sigma_1 &= (1,6)(2)(3)(4,7,5),\\
\sigma_2 &= (1,2,3,5)(4,7,6).
\end{align*}
It is easy to see that the product
\[ \pi = \sigma_1 \sigma_2 \]
is the aforementioned permutation $\pi$ which gives the clockwise order of the
labels on the polygons from $\mathcal{P}$. In our example
\[ \pi=\sigma_1 \sigma_2 = (1,2,3,4,5,6,7), \]
see~\cref{fig:polygon-oriented}.
%\medskip
With this in mind, it is easy to see that, for a given integer partition $\pi$,
the \emph{summation over oriented maps with face-type $\pi$} is equivalent to
fixing some permutation $\pi\in\Sym{|\pi|}$ with the cycle decomposition given
by the partition $\pi$, and considering all solutions to the equation
\[ \big\{ (\sigma_1,\sigma_2) : \sigma_1,\sigma_2\in \Sym{|\pi|}, \quad \sigma_1 \sigma_2 = \pi \big\}.\]
\subsection{Proof of~\texorpdfstring{\cref{thm:maps}}{Theorem 1.7}}
\label{sec:proof-oriented}
Our proof will be based on a simple double counting argument.
\begin{proof}[Proof of~\cref{thm:maps}]
In the light of the discussion from~\cref{sec:labels}, the right-hand side of
\eqref{eq:spin-stanley} can be interpreted as a sum over oriented maps with
face-type $\pi$.
% \smallskip
The oriented maps which we consider have labelled edges. We can remove these
labels and still be able to recover them if we preserve the following
information:
\begin{itemize}
\item we paint each face of the map with the colour of the corresponding
polygon from $\mathcal{P}$;
\item on each face of the map we decorate the white corner (\emph{``root''})
which corresponds to the root of the corresponding polygon from~$\mathcal{P}$;
\item on each connected component of the map we indicate the orientation.
\end{itemize}
% \smallskip
There are $2^{\ell(\pi)}$ ways to choose \emph{direction of the rotation} on
each face a map (in the spirit of~\cref{sec:non-oriented-revisited}). It follows
from~\eqref{eq:spin-stanley} that
\[ \ChSpin_\pi(\xi) = \sum_M
\frac{1}{2^{c(M)}} \frac{1}{2^{\ell(\pi)}}\ (-1)^{|\pi|-|\mathcal{V}_\circ(M)|}\ N_{M}\big( D(\xi) \big),
\]
where the sum runs over oriented maps with coloured faces, with each face having
a decorated white corner, and with each face having some selected direction of
rotation. Above, $c(M)$ denotes the number of connected components of $M$; with notations of
\eqref{eq:spin-stanley}
we clearly have $c(M)=|\langle \sigma_1,\sigma_2\rangle|$.
Let us remove the information about the orientation. In the light of
\cref{sec:non-oriented-revisited}, the resulting object is a non-oriented map
which happens to be orientable. Clearly, for each such a non-oriented but
orientable map there are $2^{c(M)}$ choices for the orientation on each
connected component of $M$. It follows that
\[ \ChSpin_\pi(\xi) = \sum_M
\frac{1}{2^{\ell(\pi)}}\ (-1)^{|\pi|-|\mathcal{V}_\circ(M)|}\ N_{M}\big( D(\xi) \big),
\]
where the sum runs over non-oriented but orientable maps with face-type~$\pi$, as required.
\end{proof}
\section{Multirectangular coordinates, spin Stanley polynomials}
\label{sec:multirectangular}
Our ultimate goal is to prove~\cref{prop:isomorphism,thm:filtration} which
provide the link between (the filtration on) the linear Kerov--Olshanski algebra
$\KO$ and its spin counterpart $\Gamma$; we will do this in
\cref{sec:proof-of-isomorphism}.
We start in this section by preparing the tools: multirectangular coordinates
and Stanley polynomials.
\begin{figure}[htb]
\begin{tikzpicture}
\clip (-0.3,-0.3) rectangle (9.5,6.5);
\fill[blue!10] (0,0) -- (7,0) -- (7,2) -- (4,2) -- (4,5) -- (0,5);
\draw[black!20] (0,0) grid (10,7);
\draw[ultra thick] (10,0) -- (0,0) -- (0,7);
\draw[thick] (7,0) -- (7,2) -- (0,2);
\draw[<->] (7.3,0) -- (7.3,2) node[midway,right] {$p_1$};
\draw[<->] (0,0.3) -- (7,0.3) node[midway,above] {$q_1$};
\draw[thick] (4,2) -- (4,5) -- (0,5);
\draw[<->] (4.3,2) -- (4.3,5) node[midway,right] {$p_2$};
\draw[<->] (0,2.3) -- (4,2.3) node[midway,above] {$q_2$};
\end{tikzpicture}
\caption{Multirectangular coordinates for partitions.}
\label{fig:multirectangular}
%\bigskip
\begin{tikzpicture}
\clip (0.7,-0.3) rectangle (10.5,6.5);
\fill[blue!10] (4,4) -- (6,4) -- (6,2)-- (8,2) -- (8,0) -- (1,0) -- (1,1) -- (2,1) -- (2,2) -- (3,2) -- (3,3) -- (4,3) -- (4,4) -- (5,4);
\begin{scope}
\clip (11,0) -- (1,0) -- (1,1) -- (2,1) -- (2,2) -- (3,2) -- (3,3) -- (4,3) -- (4,4) -- (5,4) -- (5,5) -- (6,5) -- (6,6) -- (7,6) -- (7,7) -- (11,7);
\draw[black!20] (0,0) grid (11,11);
\end{scope}
\draw[ultra thick] (10,0) -- (1,0) -- (1,1) -- (2,1) -- (2,2) -- (3,2) -- (3,3) -- (4,3) -- (4,4) -- (5,4) -- (5,5) -- (6,5) -- (6,6) -- (7,6) -- (7,7);
\draw[thick] (4,4) -- (6,4) -- (6,2) ;
\draw[thick] (2,2) -- (8,2) -- (8,0);
\draw[<->] (8.3,0) -- (8.3,2) node[midway,right] {$\proP_1$};
\draw[<->] (1,0.3) -- (8,0.3) node[midway,above] {$\proQ_1$};
\draw[<->] (6.3,2) -- (6.3,4) node[midway,right] {$\proP_2$};
\draw[<->] (3,2.3) -- (6,2.3) node[midway,above] {$\proQ_2$};
\end{tikzpicture}
\caption{Multirectangular coordinates for strict partitions.}
\label{fig:multirectangular-strict}
\end{figure}
\subsection{Multirectangular coordinates}
Following Stanley~\cite{Stanley2003/04}, for tuples of integers
$P=(p_1,\dots,p_l)$, $Q=(q_1,\dots,q_l)$ such that $p_1,\dots,p_l\geq 0$ and
$q_1\geq \cdots\geq q_l\geq 0$ we consider the corresponding
\emph{multirectangular partition} $P\times Q$, cf.~\cref{fig:multirectangular}.
De Stavola~\cite{DeStavolaThesis} adapted this notion to strict partitions: for
tuples of integers $\mathbf{P}=(\proP_1,\dots,\proP_l)$,
$\mathbf{Q}=(\proQ_1,\dots,\proQ_l)$ such that $\proP_1,\dots,\proP_l\geq 0$ and
$\proQ_2\leq \proQ_1-\proP_1$, $\proQ_3\leq \proQ_2-\proP_2$, \dots, $\proQ_l
\leq \proQ_{l-1}-\proP_{l-1}$, $0\leq \proQ_l-\proP_l$ he considered a
\emph{multirectangular strict partition} $\mathbf{P}\stimes\mathbf{Q}$,
cf.~\cref{fig:multirectangular-strict}; note that in the original work of De
Stavola both the multirectangular partition as well as the multirectangular
strict partition were denoted by the same symbol $\times$.
%\medskip
The following result gives a link between the shifted multirectangular
coordinates of a given strict partition $\xi\in\SP$ and the multirectangular
coordinates of its double $D(\xi)$.
\begin{lemma}
\label{lem:multi-strict-vs-usual}
For each $l\geq 1$ there exist polynomials
\[
\text{%
$p_1,\dots,p_l,q_1,\dots,q_l\in \Z[
\proP_1,\dots,\proP_l, \proQ_1,\dots,\proQ_l]$ of degree $1$
}
\]
with the property
that
\[ D(\mathbf{P}\stimes\mathbf{Q}) = P \times Q.\]
\end{lemma}
\begin{proof}
The desired polynomials are given by
\begin{equation}
\label{eq:double-multirectangular}
\left\{
\begin{aligned}
p_1 &= \proP_1, & q_1 &= \proQ_1+1,\\
p_2 &= \proP_2, & q_2 &= \proP_1+\proQ_2+1, \\
&\vdots & & \vdots \\
p_l &= \proP_l, & q_l &= \proP_1+\cdots+\proP_{l-1}+\proQ_l+1,\\
p_{l+1} &= \proQ_l - \proP_l, & q_{l+1} &= \proP_1+\cdots+\proP_l, \\
p_{l+2} &= \proQ_{l-1} - (\proQ_l + \proP_{l-1}), &
q_{l+2} &= \proP_1+\cdots+\proP_{l-1}, & \\
& \vdots & & \vdots \\
p_{2l} &= \proQ_1- (\proQ_2 + \proP_1), & q_{2l}&= \proP_1.
\end{aligned}
\right.
\end{equation}
\end{proof}
\subsection{Stanley polynomials}
Multirectangular coordinates $P,Q$ provide a convenient way of parametrizing
partitions; the fact that a partition can be represented in several ways with
such coordinates is not relevant.
In particular, for a function $F\in\KO$ it is convenient to consider a map
\begin{equation}
\label{eq:Stanley}
(P,Q) \mapsto F(P\times Q)
\end{equation}
which --- as it turns out --- can be identified with a unique polynomial in the
multirectangular coordinates. This polynomial, which is referred to as
\emph{Stanley polynomial}~\cite{Stanley2003/04}, is a convenient tool for
studying some enumerative and asymptotic problems of the representation theory
of the symmetric groups~\cite{DolegaFeraySniady2008}.
The explicit form of this Stanley polynomial $\Ch_\pi(P\times Q)$ in the special
case when $F=\Ch_\pi$ is the normalized character was conjectured by Stanley
\cite{Stanley-preprint2006} and proved by F\'eray~\cite{Feray2010}, see also
\cite{FeraySniady2011a}; we will refer to this result as \emph{Stanley character
formula} (it is just a reformuation of~\cref{thm:stanley-linear}).
%\medskip
De Stavola~\cite{DeStavolaThesis} initiated investigation of the analogous
coordinates for a function $F\in\Gamma$ and he asked for a closed formula for
$\ChSpin_\pi(\mathbf{P}\stimes\mathbf{Q})$.
\subsection{Closed formula for spin Stanley polynomial}
\label{sec:Stanley-strict}
Multirectangular coordinates are very convenient in the context of Stanley character formula.
More specifically, it is easy to see that
\begin{equation}
\label{eq:explicit-stanley}
N_{\sigma_1,\sigma_2}(P\times Q) =
\sum_{\kappa_2}
\prod_{c_1\in C(\sigma_1)} q_{\kappa_1(c_1)}
\prod_{c_2\in C(\sigma_2)} p_{\kappa_2(c_2)},
\end{equation}
where the sum runs over all functions
$\kappa_2\colon C(\sigma_2) \to [l]$
and the function $\kappa_1\colon C(\sigma_1)\to[l]$ is defined by
\[ \kappa_1(c_1)= \max\big\{ \kappa_2(c_2) :
c_2\in C(\sigma_2)\text{ and } c_1\cap c_2\neq \emptyset \big\}\quad \text{for } c_1\in C(\sigma_1).\]
The explicit form of the Stanley polynomial
$\ChSpin_\pi(\mathbf{P}\stimes\mathbf{Q})$ for $\pi\in\OP$ can be found by
combining~\cref{thm:spin-Stanley} applied to $\xi=\mathbf{P}\stimes\mathbf{Q}$ and
$P\times Q=D(\mathbf{P}\stimes\mathbf{Q})$ with
\cref{lem:multi-strict-vs-usual}. In order to prove the uniqueness of this
polynomial one can adapt the corresponding part of the proof of~\cite[Lemma~2.4]{DolegaFeraySniady2014}.
\section{Asymptotics of spin Stanley polynomials}
\label{sec:asymptotic-stanley}
Recall that our ultimate goal is to prove~\cref{prop:isomorphism,thm:filtration} which
provide the link between (the filtration on) the linear Kerov--Olshanski algebra
$\KO$ and its spin counterpart $\Gamma$.
As an intermediate step we present in this section the link between the
filtration on the (linear or spin) Kerov--Olshanski algebra and the degrees of
Stanley polynomials
(\cref{prop:filtration-and-degree-linear,prop:filtration-and-degree-spin}) which
might be of independent interest.
\subsection{Filtration vs Stanley polynomials: the linear case}
\begin{lemma}
\label{lem:top-degree-spin-Stanley-linear}
For arbitrary integer partition $\pi$ the corresponding linear Stanley
polynomial $\Ch_\pi(P\times Q)$ is of degree
$|\pi|+\ell(\pi)$.
Its homogeneous top-degree part is given by
\begin{equation}
\label{eq:homogeneous-top-degree-stanley-linear}
\left(\Ch_\pi\right)^{\ttop}(P\times Q):=
\sum_{\substack{\sigma_1,\sigma_2\in\Sym{k} \\ \sigma_1 \sigma_2=\pi
\\ |C(\sigma_1)|+|C(\sigma_2)|=|\pi|+\ell(\pi)}}
(-1)^{\sigma_1}\ N_{\sigma_1,\sigma_2}(P\times Q).
\end{equation}
\end{lemma}
\begin{proof}
Our strategy is to investigate the summands on the right-hand side of the
linear Stanley character formula~\eqref{eq:stanley-linear} . Note that
\eqref{eq:explicit-stanley} implies that
\begin{equation}
\label{eq:embedding-homogeneous}
N_{\sigma_1,\sigma_2}\big( P \times Q \big) \in \Z[p_1,\dots,p_l,q_1,\dots,q_l]
\end{equation}
is a homogeneous polynomial of degree $|C(\sigma_1)|+|C(\sigma_2)|$.
For a permutation
$\pi\in\Sym{k}$ we denote by $\|\pi\|:=k - |C(\pi)|$ the minimal number of
factors necessary to write $\pi$ as a product of transpositions. Triangle
inequality for $\pi=\sigma_1 \sigma_2$ implies that
\begin{equation}
\label{eq:triangle}
|C(\sigma_1)|+|C(\sigma_2)| = 2k - \big( \|\sigma_1| + \|\sigma_2\| \big) \leq 2k - \| \pi \| = |\pi| + \ell(\pi).
\end{equation}
It follows therefore that
\[ \Ch_\pi(P\times Q) \]
is indeed a polynomial of degree bounded from above by $|\pi| + \ell(\pi)$, as required.
In order to find the homogeneous top-degree part of this polynomial it is enough
to restrict the summation in~\eqref{eq:spin-stanley} to the pairs
$\sigma_1,\sigma_2$ for which~\eqref{eq:triangle} becomes equality.
\end{proof}
\begin{prop}
\label{prop:filtration-and-degree-linear}
Let $F\in \KO$ and $k\geq 0$ be an integer.
Then $F\in \F_k$ if and only if for each integer $l\geq 1$
\begin{equation}
\label{eq:my-stanley}
F(P\times Q) \in \Cbb[p_1,\dots,p_l,q_1,\dots,q_l]
\end{equation}
is a polynomial of degree at most $k$.
\end{prop}
\begin{proof}
The vector space $\F_k$ is a linear span of certain normalized characters; for
this reason it is enough to consider $F=\Ch_\pi\in\F_k$ with $\pi$ as in
\eqref{eq:filtration}.
Application of~\cref{lem:top-degree-spin-Stanley-linear} concludes the proof
of the implication in one direction.
% \medskip
We shall prove the opposite implication.
Let $m\geq 0$ be the minimal integer with the
property that $F\in\F_m$; our goal is to show that $m\leq k$. Suppose that this
is not the case and $m>k$.
We consider the map
\begin{equation}
\label{eq:homogeneous}
\Psi_m \colon \F_m \ni G \mapsto \left. \Big( [\text{homogeneous part of degree $m$}]\
G(P\times Q) \Big) \right|_{p_1=\cdots=p_l=1}
\end{equation}
which selects the top-degree homogeneous part of Stanley's polynomial and
\emph{afterwards} substitutes $p_1=\cdots=p_l=1$. We will show that $\ker
\Psi_m = \F_{m-1}$, provided $l\geq m$; this would conclude the proof because
$m>k$ implies that $F\in \ker \Psi_m = \F_{m-1}$ which contradicts the
minimality of $m$.
% \medskip
Clearly the quotient space $\F_{m}/\F_{m-1}$ is a linear span of
\begin{equation}
\label{eq:quotient}
\big(\Ch_\pi + \F_{m-1} : |\pi|+\ell(\pi)=m \big).
\end{equation}
The map $\Psi_m$ maps the family~\eqref{eq:quotient} to
\begin{equation}
\label{eq:quotient2}
\big(\Psi_m (\Ch_\pi) : |\pi|+\ell(\pi)=m \big);
\end{equation}
for our purposes of proving $\ker \Psi_m=\F_{m-1}$ it is enough to show that
\eqref{eq:quotient2} is a family of linearly independent vectors. We shall do
it in the following.
% \smallskip
In order to find the top-degree part of the polynomial
\begin{equation}
\label{eq:substitute}
\Psi_m (\Ch_\pi)\in\Z[q_1,\dots,q_l]
\end{equation}
we combine~\eqref{eq:homogeneous-top-degree-stanley-linear} with
\eqref{eq:explicit-stanley}. This top-degree part corresponds to these summands
in~\eqref{eq:homogeneous-top-degree-stanley-linear} for which $|C(\sigma_1)|$
takes the maximal possible value. This is clearly the summand for
$\sigma_1=\operatorname{id}\in\Sym{|\pi|}$ and $\sigma_2=\pi\in\Sym{|\pi|}$. In
this way we proved that~\eqref{eq:substitute} is a polynomial of degree $|\pi|$
and its homogeneous part of this maximal degree is equal to the power-sum
symmetric function
\begin{equation}
\label{eq:power-sum} p_\pi(q_1,\dots,q_l).
\end{equation}
Suppose that
\begin{equation}
\label{eq:combination} \sum_{|\pi|+\ell(\pi)=m} a_\pi\ \Psi_m (\Ch_\pi) = 0
\end{equation}
for some coefficients $a_\pi\in\Cbb$ which are not all equal to
zero and denote $n:=\max \{ |\pi| : a_\pi\neq 0 \}$. We consider the homogeneous
part of the polynomial on the left-hand side of~\eqref{eq:combination} of degree
$n$; by maximality of $n$, only partitions $\pi$ with $|\pi|=n$ contribute, and
each such a partition contributes with the power-sum symmetric function
\eqref{eq:power-sum}. Since power-sum symmetric polynomials in $l$ variables of
degree $n\leq m\leq l$ are linearly independent, it follows that $a_\pi=0$ for
$|\pi|=n$ which contradicts the definition of $n$.
This concludes the proof of the equality $\ker \Psi_m=\F_{m-1}$.
\end{proof}
This result has a spin counterpart (\cref{prop:filtration-and-degree-spin}),
however adapting the above proof to the spin setup requires
some preparations which we present below.
\subsection{Overlap double of a strict partition}
\label{sec:symmetric-double}
\begin{figure}[htb]
\begin{tikzpicture}[xscale=0.6,yscale=0.6]
\begin{scope}
\clip (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0);
\draw[gray] (0,0) grid (8,3);
\end{scope}
\begin{scope}[rotate=90,yscale=-1]
\clip (0,0) -- (0,1) -- (1,1) -- (1,2) -- (2,2) -- (2,3) -- (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0);
\draw[gray] (0,0) grid (8,3);
\end{scope}
\begin{scope}
\draw[ultra thick] (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0) -- (0,0);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (0.5,0.5) -- (5.5,0.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (1.5,1.5) -- (5.5,1.5);
\draw[line width=0.2cm, opacity=0.4,blue,line cap=round] (2.5,2.5) -- (3.5,2.5);
\end{scope}
\begin{scope}[rotate=90,yscale=-1]
\draw[ultra thick] (3,3) -- (4,3) -- (4,2) -- (6,2) -- (6,1) -- (6,0) -- (0,0);
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (0.5,0.5) -- (5.5,0.5);
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (1.5,1.5) -- (5.5,1.5);
\draw[line width=0.2cm, opacity=0.4,OliveGreen,line cap=round] (2.5,2.5) -- (3.5,2.5);
\end{scope}
\end{tikzpicture}
\caption{The overlap double $\Dover(\xi)=(6,6,4,3,2,2)$ of the strict
partition $\xi=(6,5,2)$ from~\cref{fig:strict}.}
\label{fig:doubleS}
\end{figure}
For a strict partition $\xi\in\SP$ we consider its \emph{overlap double}
$\Dover(\xi)\in \Part$ which is obtained by combining the shifted Young diagram
$\xi$ and its transpose so that they overlap along the diagonal boxes,
cf.~\cref{fig:doubleS}.
Let $p_1,\dots,p_l,q_1,\dots,q_l\in \Z[ \proP_1,\dots,\proP_l,
\proQ_1,\dots,\proQ_l]$ be the homogeneous part of degree $1$ of the polynomials
\eqref{eq:double-multirectangular}; in other words
\begin{equation}
\label{eq:double-multirectangular-H}
\left\{
\begin{aligned}
p_1 &= \proP_1, & q_1 &= \proQ_1,\\
p_2 &= \proP_2, & q_2 &= \proP_1+\proQ_2, \\
&\vdots & & \vdots \\
p_l &= \proP_l, & q_l &= \proP_1+\cdots+\proP_{l-1}+\proQ_l,\\
p_{l+1} &= \proQ_l - \proP_l, & q_{l+1} &= \proP_1+\cdots+\proP_l, \\
p_{l+2} &= \proQ_{l-1} - (\proQ_l + \proP_{l-1}), &
q_{l+2} &= \proP_1+\cdots+\proP_{l-1}, & \\
& \vdots & & \vdots \\
p_{2l} &= \proQ_1- (\proQ_2 + \proP_1), & q_{2l}&= \proP_1.
\end{aligned}
\right.
\end{equation}
It is easy to check that with this choice
\begin{equation}
\label{eq:symmetric-double-multi}
\Dover(\mathbf{P}\stimes\mathbf{Q}) = P \times Q.
\end{equation}
\subsection{Top-degree part of spin Stanley polynomials}
\cref{lem:top-degree-spin-Stanley-linear} has the following spin counterpart.
\begin{lemma}
\label{lem:top-degree-spin-Stanley}
For arbitrary $\pi\in\OP_k$ the corresponding spin Stanley polynomial
$\ChSpin_\pi(\mathbf{P}\stimes\mathbf{Q})$ is of degree $|\pi|+\ell(\pi)$. Its
homogeneous top-degree part is given by
\begin{multline}
\label{eq:homogeneous-top-degree-stanley}
\left(\ChSpin_\pi\right)^{\ttop}(\mathbf{P}\stimes\mathbf{Q})
\\
\coloneqq \sum_{\substack{\sigma_1,\sigma_2\in\Sym{k} \\ \sigma_1 \sigma_2=\pi \\ |C(\sigma_1)|+|C(\sigma_2)|=|\pi|+\ell(\pi)}}
\frac{1}{2^{|\sigma_1\vee \sigma_2|}}\ (-1)^{\sigma_1}\ N_{\sigma_1,\sigma_2}\big( \Dover(\mathbf{P} \stimes\mathbf{Q}) \big).
\end{multline}
\end{lemma}
\begin{proof}
We revisit the proof of~\cref{lem:top-degree-spin-Stanley-linear} but this time
we consider the spin Stanley formula~\eqref{eq:spin-stanley}.
\cref{lem:multi-strict-vs-usual} implies therefore that
\begin{equation}
\label{eq:embedding-inhomogeneous}
N_{\sigma_1,\sigma_2}\big( D(\mathbf{P} \stimes\mathbf{Q}) \big) \in \Z[\proP_1,\dots,\proP_l,\proQ_1,\dots,\proQ_l]
\end{equation}
is an \emph{inhomogeneous} polynomial of the degree $|C(\sigma_1)|+|C(\sigma_2)|$, obtained from
\eqref{eq:embedding-homogeneous} by the substitution
\eqref{eq:double-multirectangular}.
The homogeneous part of~\eqref{eq:embedding-inhomogeneous} of this top degree
can therefore be obtained from~\eqref{eq:embedding-homogeneous} by the
\emph{homogeneous} substitution~\eqref{eq:double-multirectangular-H}. Equation
\eqref{eq:symmetric-double-multi} implies therefore that this homogeneous part
is equal to
\begin{equation}
\label{eq:embedding-homogeneous-again}
N_{\sigma_1,\sigma_2}\big( \Dover(\mathbf{P} \stimes\mathbf{Q}) \big) \in \Z[\proP_1,\dots,\proP_l,\proQ_1,\dots,\proQ_l].
\end{equation}
%\medskip
Inequality~\eqref{eq:triangle} implies therefore that
\[ \ChSpin_\pi(\mathbf{P}\stimes\mathbf{Q}) \]
is indeed a polynomial of degree bounded from above by $|\pi| + \ell(\pi)$, as required.
In order to find the homogeneous top-degree part of this polynomial it is enough
to: (a) restrict the summation in~\eqref{eq:spin-stanley} to pairs
$\sigma_1,\sigma_2$ for which~\eqref{eq:triangle} becomes equality and then (b)
to consider the homogeneous part~\eqref{eq:embedding-homogeneous-again} of the
surviving summands. This concludes the proof.
\end{proof}
\subsection{Filtration vs Stanley polynomials: the spin case}
\begin{prop}
\label{prop:filtration-and-degree-spin}
Let $F\in \Gamma$ and $k\geq 0$ be an integer.
Then $F\in \Gcal_k$ if and only if for each integer $l\geq 1$
\[ F(\mathbf{P}\stimes\mathbf{Q}) \in \Cbb[\proP_1,\dots,\proP_l,\proQ_1,\dots,\proQ_l] \]
is a polynomial of degree at most $k$.
\end{prop}
\begin{proof}
We revisit the proof of~\cref{prop:filtration-and-degree-linear} and review the
necessary changes.
We replace each occurrence of $\F_k$ by $\Gcal_k$ and each linear character
$\Ch_\pi$ with $\pi\in\Part$ by its spin counterpart $\ChSpin_\pi$ with
$\pi\in\OP$. Reference to~\cref{lem:top-degree-spin-Stanley-linear} should be
replaced by~\cref{lem:top-degree-spin-Stanley}.
Instead of $\Psi_m$ we consider the map
\begin{equation}
\label{eq:homogeneous-spin}
\Psi^{\text{spin}}_m \colon \Gcal_m \ni G \mapsto \left. \Big( [\text{homogeneous part of degree $m$}]\
G(\mathbf{P}\stimes\mathbf{Q}) \Big) \right|_{\proP_1=\cdots=\proP_l=1}
\end{equation}
which selects the top-degree homogeneous part of spin Stanley's polynomial and
\emph{afterwards} substitutes $\proP_1=\cdots=\proP_l=1$.
%\medskip
The only part which requires some changes in the spin context is the proof that
if $m=|\pi|+\ell(\pi)$ then
$\Psi^{\text{spin}}_m(\ChSpin_\pi)\in\Q[\proQ_1,\dots,\proQ_l]$ is a polynomial
of degree $|\pi|$ and its homogeneous part of this maximal degree is equal to
the power-sum symmetric function $p_\pi(\proQ_1,\dots,\proQ_l)$. We present this
proof below.
%\smallskip
Let us fix some $\sigma_1,\sigma_2\in\Sym{|\pi|}$ which contribute to
\eqref{eq:homogeneous-top-degree-stanley}. The colorings which contribute to
$N_{\sigma_1,\sigma_2}(\lambda)$ with
\[\lambda=\Dover\big( (1,\dots,1) \stimes
\mathbf{Q} \big)= \Dover(\proQ_1,\dots,\proQ_l)\]
can be enumerated by the following algorithm. Firstly, we select some set of
cycles of $\sigma_1$ and some set of cycles of $\sigma_2$ (we will call the
cycles which belong to them \emph{special cycles}). Secondly, we assign to these
special cycles the values of $f_1$ and $f_2$ from the set $[l]$ in an arbitrary
way. The above two steps do not depend on the variables $\proQ_1,\dots,\proQ_l$,
but they do depend on $l$, the number of the variables.
In the third step we associate to all non-special cycles the values of $f_1$ and
$f_2$ from the set $\{l+1,l+2,\dots\}$ in such a way that $f=(f_1,f_2)$ is a
colouring of $(\sigma_1,\sigma_2)$ which is compatible with $\lambda$. Without
loss of generality we may assume that whenever two cycles $c_1\in C(\sigma_1)$,
$c_2\in C(\sigma_2)$ intersect, at least one of them is special; otherwise the
column $f_1(c_1)$ and the row $f_2(c_2)$ would intersect outside of the Young
diagram $\lambda$ and there would be no such colourings which are compatible
with $\lambda$. For each $i\in [l]$ in $i$-th row of $\lambda$ there are
$\proQ_{i}+i-1$ boxes and $l$ of them are forbidden by non-speciality
requirement; it follows that the number of choices in this third step is equal
to
\begin{equation}
\label{eq:number-of-choices}
\left(\prod_{\substack{c_1 \in C(\sigma_1), \\ \text{$c_1$ is not special} }} \Mk \left[ \proQ_{g_1(c_1)} \Mk +\Mk g_1(c_1)\Mk- \Mk 1 \Mk - \Mk l \right] \right) \Mk \left(
\prod_{\substack{c_2 \in C(\sigma_2), \\ \text{$c_2$ is not special} }}
\left[ \proQ_{g_2(c_2)} \Mk+\Mk g_2(c_2)\Mk-\Mk 1\Mk- l \right] \right),
\end{equation}
where for $c_1\in C(\sigma_1)$, $c_2\in C(\sigma_2)$ which are not special we define
\begin{align*}
g_1(c_1) &= \max \big\{ f_2(c_2) : c_2\in C(\sigma_2), \text{$c_2$ is special} \\ & \qquad\qquad\qquad\qquad \text{and the cycles $c_1$ and $c_2$ intersect} \big\},\\
g_2(c_2) &= \max \big\{ f_1(c_1) : c_1\in C(\sigma_1), \text{$c_1$ is special} \\ & \qquad\qquad\qquad\qquad \text{and the cycles $c_1$ and $c_2$ intersect} \big\}.
\end{align*}
%\smallskip
The exact form of~\eqref{eq:number-of-choices} is not really important; we will
use only the observation that it is a polynomial in $\proQ_1,\dots,\proQ_l$ of
degree equal to the number of non-special cycles. Since we are interested in the
top-degree part of the polynomial
$\Psi^{\text{spin}}_m(\ChSpin_\pi) \in\Q[\proQ_1,\dots,\proQ_l]$, hence our goal is
to maximize the number of such non-special cycles.
%\smallskip
In the following it will be sometimes convenient to identify a given cycle $c\in
C(\sigma_1)\sqcup C(\sigma_2)$ with its support $c\subseteq [|\pi|]$ which is a
non-empty subset of $[|\pi|]=\{1,\dots,|\pi|\}$. We denote by $N\subseteq
C(\sigma_1)\sqcup C(\sigma_2)$ the set of all non-special cycles. We consider an
arbitrary map $s\colon N\to[|\pi|]$ which to any non-special cycle associates
one of its elements, \ie $f(c)\in c$ for $c\in C(\sigma_1)\sqcup C(\sigma_2)$.
The map $s$ is injective; otherwise this would contradict the assumption that
any two non-special cycles are disjoint. This has twofold consequences.
Firstly, the number of non-special cycles is bounded from above by
$|\pi|$ so the degree of the polynomial
$N_{\sigma_1,\sigma_2}(\lambda)\in\Z[\proQ_1,\dots,\proQ_l]$ is bounded from
above by $|\pi|$.
Secondly, if this bound on the number of non-special cycles is saturated, the
map $s$ is a bijection. In this case for each $x\in[|\pi|]$ exactly one of the
following two possibilities holds true: either
\begin{itemize}
\item $(x)$ is a non-special cycle
of $\sigma_1$ and $x$ belongs to some special cycle of $\sigma_2$, or
\item $(x)$ is
a non-special cycle of $\sigma_2$ and $x$ belongs to some special cycle of
$\sigma_1$.
\end{itemize}
The above discussion implies that the product $\sigma_1 \sigma_2\in\Sym{|\pi|}$
can be calculated in a very simple way. Indeed, let us consider some special
cycle $c$; let us say that it is a cycle of $\sigma_1$. Since $\sigma_2$
restricted to the support of $c$ is equal to the identity, the restriction of
the product $\sigma_1 \sigma_2$ to the support of $c$ coincides with the cycle $c$.
An analogous result remains true in the case when $c$ is a special cycle of
$\sigma_2$. In this way we proved that the product $\sigma_1\sigma_2$ coincides
with the collection of the special cycles.
%\smallskip
In this way we proved that if the number of non-special cycles achieves its
maximal value $|\pi|$, each cycle of $\pi=\sigma_1\sigma_2\in\Sym{|\pi|}$ is
either a special cycle of $\sigma_1$ or a special cycle of $\sigma_2$ and there
are no other special cycles.
For example, if $\pi=(\pi_1)\in\OP$ is an odd partition which consists of only
one part and
\[ \pi=(1,2,\dots,\pi_1)\in\Sym{\pi_1}\]
is the corresponding permutation, the maximal number of non-special cycles is
obtained for the following two choices:
\begin{align*}
\sigma_1&=\id=\underbrace{(1)(2)\cdots(\pi_1)}_{\text{non-special cycles}}, & \sigma_2&=\underbrace{(1,2,\dots,\pi_1)}_{\text{special cycle}}, \\
\intertext{and}
\sigma_1&=\underbrace{(1,2,\dots,\pi_1)}_{\text{special cycle}}, &
\sigma_2&=
\id=\underbrace{(1)(2)\cdots(\pi_1)}_{\text{non-special cycles}}.
\end{align*}
If $j$ denotes the value of the function $f_i$, $i\in\{1,2\}$, on the unique
special cycle, the homogeneous top-degree part of the corresponding summand
\[ \frac{1}{2^{|\sigma_1\vee \sigma_2|}}\ (-1)^{\sigma_1}\ N_{\sigma_1,\sigma_2}(\lambda) \]
on the right-hand side of~\eqref{eq:homogeneous-top-degree-stanley} is equal to
\[ \frac{1}{2} \sum_j \proQ_j^{\pi_1} \]
because $\pi_1$ is an odd integer.
In general, for each cycle of $\pi$ there are two choices analogous to the ones above.
It follows that in total there are $2^{\ell(\pi)}$ choices for $\sigma_1$ and
$\sigma_2$ and which of the cycles are special.
For each such a choice the total
contribution of~\eqref{eq:number-of-choices} is a polynomial of degree $|\pi|$,
with the homogeneous part equal to $\frac{1}{2^{\ell(\pi)}}
p_{\pi}(\proQ_1,\dots,\proQ_l)$, a multiple of the power-sum symmetric
polynomial. This concludes the proof.
\end{proof}
\section{Proof of~\texorpdfstring{\cref{prop:isomorphism,thm:filtration}}{Theorems 1.8 and 1.9}}
\label{sec:proof-of-isomorphism}
\label{sec:proof-of-filtration}
\begin{proof}[Proof of~\cref{prop:isomorphism}]
\
\emph{Proof that $\double(\KO)\subseteq\Gamma$.}
The algebra $\KO$ has an algebraic basis given by the functions $S_2,S_3,\dots$ with
\[ S_k(\lambda) = (k-1) \iint_{(x,y)\in\lambda} (x-y)^{k-2} \dif x \ \dif y,
\]
see~\cite{DolegaFeraySniady2008}.
The $i$-th row of a shifted Young diagram $\xi$ corresponds to two rectangles
in the double $D(\xi)$. By calculating the integral over each of the rectangles
it follows that
\begin{multline*}
(\double S_k)(\xi) = \\
\frac{1}{k} \sum_{i} \left( - \xi_i^k + (\xi_i+1)^k + 0^k - 1^k \right) + \left( -(1-\xi_i)^k + (-\xi_i)^k +1^k - 0^k \right).
\end{multline*}
It is easy to check that the polynomial in the variable $\xi_i$ which appears
in the above formula is an odd polynomial; it follows therefore that $\double
S_k\in \Cbb[p_1,p_3,p_5,\dots]$ is a symmetric function which is supersymmetric.
By a result of Ivanov~\cite[Section~6]{Ivanov2004} it follows that $\double S_k\in \Gamma$.
% \medskip
\emph{Proof that $\double$ is surjective.} We will
prove a stronger result $\Gcal_k \subseteq \double(\F_k)$ below, in the proof of
\cref{thm:filtration}.
\end{proof}
\begin{proof}[Proof of~\cref{thm:filtration}]
\
\emph{Proof that the family $(\Gcal_k)$ is a filtration on the algebra $\Gamma$.}
Suppose that $F_1\in \Gcal_{k_1}$ and $F_2\in \Gcal_{k_2}$. We apply
\cref{prop:filtration-and-degree-spin}; for $i\in\{1,2\}$ it follows that
$F_i(\mathbf{P}\stimes\mathbf{Q})\in\Cbb[\proP_1,\dots,\proP_l,\proQ_1,\dots,\proQ_l]$
is a polynomial of degree at most $k_i$ hence
the product $F_1(\mathbf{P}\stimes\mathbf{Q}) F_2(\mathbf{P}\stimes\mathbf{Q})$
is a polynomial of degree at most $k_1+k_2$. We apply
\cref{prop:filtration-and-degree-spin} again; it follows that $F_1 F_2\in
\Gcal_{k_1+k_2}$. This implies that $(\Gcal_k)$ is indeed a filtration.
%\medskip
\emph{Proof of the inclusion $\double(\F_k) \subseteq \Gcal_k$.} Suppose $F\in\F_k$. By
\cref{prop:filtration-and-degree-linear} and~\cref{lem:multi-strict-vs-usual} it
follows that $\double(F)(\mathbf{P}\stimes\mathbf{Q})$ is a polynomial of degree
at most $k$. We apply~\cref{prop:filtration-and-degree-spin}; it follows that
$\double(F)\in \Gcal_k$, as required.
%\medskip
\emph{Proof of the inclusion $\Gcal_k \subseteq \double(\F_k)$.} Let us fix an integer $k\geq
0$. Let $\pi\in\OP$ be an arbitrary odd partition such that $|\pi|+\ell(\pi)\leq
k$. We define
\[ x_\pi:= - \sum_{\partition} \left(-\frac{1}{2}\right)^{|\partition|}\ (2|\partition|-3)!!\
\prod_{b\in \partition} \Ch_{(\class_i:i \in b)} \in \F_k,
\]
where the sum runs over all set-partitions of the set $[\ell(\class)]$.
\cref{theo:spin-in-linear} implies that $\double\left(x_\pi\right)=\ChSpin_\pi$;
in this way we proved that $\ChSpin_\pi\in \double(\F_k)$. From the definition
\eqref{eq:filtration2} of $\Gcal_k$ it follows that $\Gcal_k \subseteq \double(\F_k)$,
as required.
\end{proof}
\longthanks{We thank Valentin F{\'e}ray and Maciej Do\l\k{e}ga for several inspiring discussions.}
\bibliographystyle{amsplain-ac}
\bibliography{ALCO_Matsumoto_253}
\end{document}