%% if you are submitting an initial manuscript then you should have submission as an option here
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\documentclass[finalversion]{FPSAC2021}
\articlenumber{84}
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\usepackage{mathrsfs,latexsym,mathtools,mathdots,booktabs,enumerate,tikz,tikz-cd,bm,url}
\usepackage[centertableaux]{ytableau}
\usepackage[utf8]{inputenc}

\usetikzlibrary{arrows,positioning,arrows.meta,shapes,decorations.markings,matrix}
\usepackage[capitalize,noabbrev]{cleveref}
\crefname{ineq}{inequality}{inequalities}
\creflabelformat{ineq}{#2{\upshape(#1)}#3}

%% define your title in the usual way
\title[Skew characters and cyclic sieving]{Skew characters and cyclic sieving\\ (extended abstract)}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{%
Per Alexandersson\thanks{\href{mailto:per.w.alexandersson@gmail.com}{per.w.alexandersson@gmail.com}. %
Per Alexandersson was supported by \emph{Swedish Research Council (Vetenskapsrådet), grant 2015-05308}.}\addressmark{1},
Stephan Pfannerer\thanks{\href{mailto:stephan.pfannerer@tuwien.ac.at}{stephan.pfannerer@tuwien.ac.at}. Stephan Pfannerer was partially supported by  the  Austrian  Science  Fund(FWF) P29275 and is a recipient of a DOC Fellowship of the Austrian Academy of Sciences.}\addressmark{2},
Martin Rubey\thanks{\href{mailto:martin.rubey@tuwien.ac.at}{martin.rubey@tuwien.ac.at}. Martin Rubey was partially supported by  the  Austrian  Science  Fund(FWF) P29275.}\addressmark{2}, \and \\
\ Joakim Uhlin\thanks{\href{mailto:joakim\_uhlin@hotmail.com}{joakim\_uhlin@hotmail.com}.}\addressmark{1}%
}

%% then use \addressmark to match authors to institutions here
\address{%
\addressmark{1}Department of Mathematics, Stockholm University, S-10691, Stockholm, Sweden
\addressmark{2}Fakult\"at f\"ur Mathematik und Geoinformation, TU Wien, Austria
% \\ \addressmark{2}School of Something, University of Something Else, Somewhere
}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
  In 2010, B.~Rhoades proved that promotion on rectangular standard
  Young tableaux together with the associated fake-degree polynomial
  shifted by an appropriate power,
  provides an instance of the cyclic sieving phenomenon.

  Motivated in part by this result, we show that we can expect a cyclic
  sieving phenomenon for $m$-tuples of skew standard Young tableaux
  of the same shape and the $m^\textnormal{th}$ power of the
  associated fake-degree polynomial, for fixed $m$, under mild and
  easily checked conditions.  However, we are unable to exhibit an
  appropriate group action explicitly.

  Put differently, we determine in which cases the $m^\textnormal{th}$ tensor
  power of a skew character of the symmetric group carries a
  permutation representation of the cyclic group.

  To do so, we use a method proposed by N.~Amini and the first
  author, which amounts to establishing a bound on the number of
  border-strip tableaux of skew shape.

  Finally, we apply our results to the invariant theory of tensor
  powers of the adjoint representation of the general linear group.
  In particular, we prove the existence of a bijection between
  permutations and J.~Stembridge's alternating tableaux, which
  intertwines rotation and promotion.
}

%% put your French abstract here, or comment this out if you don't have one
% \resume{parlez vous fran\c{c}ais?}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Cyclic sieving, Skew characters, Border strip tableaux, Adjoint representation}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
% \bibliography{bibliography}
\addbibresource{84Alexandersson.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\usepackage{subcaption}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

\newtheorem*{note}{Note}

\newcommand{\defin}[1]{\textcolor{blue}{\emph{#1}}}

\newcommand{\setNzero}{\mathbb{N}_0}
\newcommand{\setNpos}{\mathbb{N}_{\geq 1}}
\newcommand{\setZ}{\mathbb{Z}}
\newcommand{\setC}{\mathbb{C}}

\newcommand{\xvec}{\mathbf{x}}
\newcommand{\yvec}{\mathbf{y}}

\newcommand{\powerSum}{\mathrm{p}}
\newcommand{\monomial}{\mathrm{m}}
\newcommand{\schurS}{\mathrm{s}}

\newcommand{\thsup}{\textnormal{th}}
\newcommand{\rdsup}{\textnormal{rd}}

%Used with Fomin-Lulov.
\newcommand{\fbound}{B}

\newcommand{\BSTtoSYT}{\mathcal{I}}

% Tableaux and sets
\newcommand{\SSYT}{\mathrm{SSYT}}
\newcommand{\SYT}{\mathrm{SYT}}
\newcommand{\SYTT}{\mathrm{SYTT}}
\newcommand{\BST}{\mathrm{BST}}

\newcommand{\COF}{\mathrm{COF}}

% groups etc.
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\symS}{\mathfrak{S}}
\newcommand{\gl}{\mathfrak{gl}}


% Last section notation
\DeclareMathOperator{\st}{st} % the mystery statistic
\DeclareMathOperator{\shape}{sh}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\newcommand{\Perm}{\mathcal{P}} % the bijection between alternating tableaux and permutations
\newcommand{\Alt}[2]{\mathcal A_{#2}^{(#1)}} % the set of alternating tableaux
\newcommand{\alt}{\mathcal{A}} % an alternating tableau
\newcommand{\wgt}{\mu} % a weight
\DeclareMathOperator{\rot}{rot} % rotation of a permutation
\DeclareMathOperator{\pr}{pr} % promotion of an alternating tableau

\def\longcycle{(1,\dots,n)}
\def\tocyclic{\!\downarrow_{\langle\longcycle\rangle}} % restriction to the long cycle

%Math operators
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\core}{core}
\DeclareMathOperator{\quotient}{quot}
\DeclareMathOperator{\height}{height}
\DeclareMathOperator{\maj}{maj}
\DeclareMathOperator{\rect}{rect}

\newcommand{\qbinom}{\genfrac{[}{]}{0pt}{}}

\tikzset{iTrio/.pic={%
		\begin{scope}[yscale=-1]
			\draw[black,x=1em,y=1em] (0,0)--(1,0)--(1,3)--(0,3)--(0,0);
		\end{scope}
}}
\tikzset{hTrio/.pic={%
		\begin{scope}[yscale=-1]
			\draw[black,x=1em,y=1em] (0,0)--(0,1)--(3,1)--(3,0)--(0,0);
		\end{scope}
}}
\tikzset{jTrio/.pic={%
		\begin{scope}[yscale=-1]
			\draw[black,x=1em,y=1em] (1,0)--(2,0)--(2,2)--(0,2)--(0,1)--(1,1)--(1,0);
		\end{scope}
}}
\tikzset{tTrio/.pic={%
		\begin{scope}[yscale=-1]
			\draw[black,x=1em,y=1em] (0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--(0,0);
		\end{scope}
}}

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
This is an extended abstract of \cite{AlexanderssonPfannererRubeyUhlin2020x}.
We determine which tensor powers of a skew character $\chi^{\lambda/\mu}$ of the symmetric group $\symS_n$ carry a permutation representation of the cyclic group of order $n$.

This problem can be rephrased in terms of V.~Reiner, D.~Stanton \&
D.~White's \emph{cyclic sieving
  phenomenon}~\cite{ReinerStantonWhite2004}.  Let $\SYT(\lambda/\mu)$
be the set of standard Young tableaux of skew shape $\lambda/\mu$,
and let $f^{\lambda/\mu}(q)$ be G.~Lusztig's fake degree polynomial
for $\chi^{\lambda/\mu}$.
%
Then there \emph{exists} an action $\rho$ of the cyclic group of
order $n=|\lambda/\mu|$ such that
\begin{equation*}
  \left(\underbrace{\SYT(\lambda/\mu)\times\dots\times\SYT(\lambda/\mu)}_m, %
    \langle \rho \rangle, %
    f^{\lambda/\mu}(q)^m\right)
\end{equation*}
exhibits the cyclic sieving phenomenon, if and only if
$f^{\lambda/\mu}(q)^m$ evaluates to nonnegative integers at
$n^\thsup$ roots of unity.\footnote{In fact, in the spirit of
  Rhoades' result~\cite{Rhoades2010}, it is easy to see that this
  equivalence also holds for $q^k f^{\lambda/\mu}(q)^m$, where $k$ is
  any integer.}
%
If $m$ is even this is always the case.  If $m$ is odd, this is the
case if and only if there exists a tiling of $\lambda/\mu$ with
border-strips of size $k$ of even height for every $k\mid n$, see
\cref{prop:positive-cyclic-sieving}.  Remarkably, a theorem of
Swanson \cite[thm.~1.5] {Swanson2018} implies that in this case, and
provided that $\mu$ is the empty partition, the cyclic group action
has a free orbit.

We also show that for any skew shape $\lambda/\mu$ and any integer $s>0$ there is an action $\tau$ of the cyclic group of order $s$ on \emph{stretched shapes} such that
\begin{equation*}
  \left(\SYT(s\lambda/s\mu), \langle \tau \rangle, %
    f^{s\lambda/s\mu}(q)\right)
\end{equation*}
exhibits the cyclic sieving phenomenon, see \cref{thm:stretchedCSPtheorem}.

At this point we are unable to present $\rho$ and $\tau$ explicitly for general skew shapes $\lambda/\mu$.
Instead, we use a characterization of P.~Alexandersson \& N.~Amini~\cite{AlexanderssonAmini2018}, which says that $f\in\setNzero[q]$ is a cyclic sieving polynomial for a group action of the cyclic group of order $n$, if and only if for a primitive $n^\thsup$ root of unity $\xi$ and all $k\mid n$ we have that $f(\xi^k)\in\setNzero$ and
\begin{equation*}
\sum_{d\mid k}\mu(k/d)f(\xi^d)\geq 0,
\end{equation*}
where $\mu$ is the number-theoretic M\"obius function.

To apply this result, we establish a new bound for the absolute value of the skew
character evaluated at a power of the long cycle.  More precisely, \cref{thm:BST-bound-skew} implies that, for any $k\mid n$,
\begin{equation*}
  |f^{\lambda/\mu}(\xi^k)| \ge \sum_{d\mid k, d<k} |f^{\lambda/\mu}(\xi^d)|
\end{equation*}
provided $|f^{\lambda/\mu}(\xi^k)|\ge 2$.

To prove this inequality, we note that
$|f^{\lambda/\mu}(\xi^d)| = |\chi^{\lambda/\mu}((m^d))| =
|\BST(\lambda/\mu,m)|$,
the number of border-strip tableaux of shape $\lambda/\mu$ with
strips of size $m$, extending the theorems for partitions by
T.~Springer~\cite{Springer1974} and G.~James \&
A.~Kerber~\cite{JamesKerber1984}.  We then approximate the number of
border-strip tableaux using a bound by S.~Fomin \&
N.~Lulov~\cite{FominLulov1997}.

Our main motivation is an implication for the invariant theory of the
general linear group, as we now explain.  Let $\gl_r$ be the adjoint
representation of $\GL_r$, and consider its $n^\thsup$ tensor power
$\gl_r^{\otimes n}$.  The symmetric group $\symS_n$ acts on this space
by permuting tensor positions.  Thus, using Schur--Weyl duality, we
can determine the subspace of $\GL_n$-invariants of
$\gl_r^{\otimes n}$, regarded as a representation of $\symS_n$.  This representation
turns out to be isomorphic to
\[
  \bigoplus_{\substack{\lambda\vdash n\\\ell(\lambda)\leq r}} S_\lambda\otimes S_\lambda,
\]
where the direct sum is over all partitions of $n$ into at most $r$
parts, and $S_\lambda$ is the irreducible representation of $\symS_n$
corresponding to $\lambda$.  In particular, for $r\geq n$, the
dimension of the space of invariants equals the size of $\symS_n$.

A fundamental question of invariant theory is to find an explicit
basis of the space of invariants, which, if possible, enjoys further
desirable properties.  One such property is invariance under rotation
of tensor positions, following G.~Kuperberg's idea of web bases~\cite{Kuperberg1996}.

An elegant and useful solution would be to describe a set of
permutations in $\symS_n$, and a bijection from these to the basis
elements which intertwines rotation of permutations (that is,
conjugation with the long cycle) and rotation of tensor positions.
It would be even nicer if this set of permutations for the invariants
of $\gl_r^{\otimes n}$ were a subset of the set of permutations
for the invariants of $\gl_{r+1}^{\otimes n}$.

Although it appears to be difficult to exhibit such an intertwining
bijection explicitly, our results, combined with previous work
of S.~Pfannerer, M.~Rubey \& B.~Westbury~\cite{PfannererRubeyWestbury2020},
implies that such a solution must exist, see \cref{thm:existence-pr-rot}.

The existence of such an intertwining bijection is closely related to the existence of a rotation invariant statistic $\st$ mapping permutations to partitions, such that $|\{\sigma\in \symS_n: \st(\sigma)=\lambda\}| = |\SYT(\lambda)\times \SYT(\lambda)|$, see \cref{cor:rotationInvatiantStat}.

\section{Cyclic group actions and cyclic sieving}\label{sec:csp}

In this section we recall V.~Reiner, D.~Stanton \& D.~White's cyclic sieving
phenomenon, characters of cyclic group actions and a result of
P.~Alexandersson \& N.~Amini characterizing characters arising from cyclic
group actions.


\begin{definition}[\cite{ReinerStantonWhite2004}]
  Let $X$ be a finite set and let $\rho$ be a generator of an
  action of the cyclic group of order $n$ on $X$.

  Given a polynomial $f(q)\in \setNzero[q]$ we say that the triple
  $(X,\langle\rho\rangle,f(q))$ \defin{exhibits the cyclic sieving
    phenomenon} if for all $d \in \setZ$
  \begin{align*}
    \#\{ x\in X : \rho^d \cdot x = x \} = f(\xi^d),
  \end{align*}
  where $\xi$ is a primitive $n^\thsup$ root of unity.  In this case
  $f(q)$ is a \defin{cyclic sieving polynomial} for the group
  action.
\end{definition}
In particular, the cardinality of $X$ is given by $f(1)$.  More
generally, realizing the cyclic group of order $n$ as the group of
$n^\thsup$ roots of unity and identifying its ring of characters with
$\setZ[q]/(q^n-1)$, the cyclic sieving polynomial $f(q)$ modulo
$q^n-1$ reduces to the character of the group action.

Much attention has been given to prove cyclic sieving phenomena on
certain families of tableaux. %, see \cref{tbl:tableaux}.
Most famously, B.~Rhoades \cite{Rhoades2010} showed that $\SYT(a^b)$,
together with promotion and the fake degree polynomial $f^\lambda(q)$
associated with $\lambda=(a^b)$, exhibits the cyclic sieving
phenomenon.

In many cases, the only known way to prove that a given triple
$(X,\langle\rho\rangle,f(q))$ exhibits the cyclic sieving phenomenon
is to enumerate the number of fixed points of the group action, and
then verify that the evaluation of the polynomial yields the same
number.  A different strategy is to exhibit a representation of the
symmetric group such that the given group action is the restriction
of the representation to the cyclic group generated by the long
cycle.

Our main result, \cref{prop:positive-cyclic-sieving}, is of a rather
different flavour.  Instead of computing the character of a given
cyclic group action, we consider a specific representation of the
symmetric group, and determine whether its restriction to the cyclic
group generated by the long cycle is isomorphic to a cyclic group
action.  Put differently, we determine whether this restriction is a
permutation representation of the cyclic group.  To do so, we use the
following criterion.
\begin{theorem}[{\cite[Thm. 2.7]{AlexanderssonAmini2018}}]\label{thm:alexanderssonAmini}
  Let $f(q)\in \setNzero[q]$ and suppose that $f(\xi^d)\in\setNzero$ for all
  $d\in\{1,\dots,n\}$, where $\xi$ is a primitive $n^\thsup$ root of
  unity.  Let $X$ be any set of size~$f(1)$.

  Then there exists a
  cyclic group action $\rho$ of order $n$ such that
  $(X,\langle\rho\rangle,f(q))$ exhibits the cyclic sieving phenomenon if and only if for
  every $k\mid n$,
  \[
  \sum_{d \mid k} \mu(k/d) f(\xi^d)\geq 0,
  \]
  where $\mu$ is the number-theoretic M\"obius function.
\end{theorem}
Note that, except for its size, the nature of the set $X$ is
irrelevant in this theorem.
\begin{remark}\label{rmk:orbit-sizes}
  If $(X,\langle\rho\rangle,f(q))$ exhibits the cyclic sieving
  phenomenon, the expression
  \[
    \sum_{d\mid k} \mu(k/d) f(\xi^d)
  \]
is the number elements in orbits of size $k$ of the group action.
  Hence, it must be nonnegative and divisible by $k$.
\end{remark}
We conclude this section by recalling a fact that makes cyclic group
actions special.  Suppose we are given a linear representation and a
group action of a finite group which are isomorphic as linear
representations.  Then, trivially, there is a basis of the invariant
space of the representation such that the restriction of the
representation to this basis is isomorphic to the group action.
However, two non-isomorphic group actions may be isomorphic as linear
representations.  This is not the case for group actions of a cyclic
group, as R.~Brauer's permutation lemma shows:
%
\begin{theorem}[\cite{Brauer1941,Kovacs1982}]\label{thm:Brauer}
  Two cyclic group actions are isomorphic if and only if they are
  isomorphic as linear representation, that is, their characters coincide.
\end{theorem}



\section{Character values and border strip tableaux}

\begin{definition}
Given a skew standard Young tableau $T$ with $n$ cells,
a label~$j$ with $1\leq j< n$ is a \defin{descent} of~$T$,
if the label $j+1$ appears in a row strictly below that of~$j$.
The \defin{major index} of $T$, denoted $\maj(T)$, is the sum of the descents of~$T$.
%
The \defin{(skew) fake-degree polynomial} $f^{\lambda/\mu}(q)$
associated with a skew Young diagram $\lambda/\mu$
is the generating function for the major index:
\[
 f^{\lambda/\mu}(q) \coloneqq \sum_{T \in \SYT(\lambda/\mu)} q^{\maj(T)}.
\]
\end{definition}
In this section we give two alternative expressions for the
fake-degree polynomial evaluated at a primitive $n^\thsup$ root of unity.
The first is an evaluation of a skew character at a power of the
\defin{long cycle} $\longcycle \in \symS_n$, generalizing a result by T.~Springer~\cite{Springer1974}.
The second one as the cardinality of a certain set of border strip tableaux times a sign.
\begin{definition}
A \defin{border-strip} (or \defin{ribbon} or \defin{skew hook})
is a connected non-empty skew Young diagram containing no $2 \times 2$-square of cells,
as in \cref{fig:borderStripExample}.
The \defin{height} of a border-strip is one less than the number of rows it spans. Its \defin{tail} is the leftmost cell in the last row of a border-strip.

Let $\lambda/\mu$ be a skew shape. The \defin{size} of $\lambda/\mu$ is its number of cells, denoted $|\lambda/\mu|$. Suppose that $\nu = (\nu_1, \dotsc, \nu_\ell)$ is a partition of $|\lambda/\mu|$. A \defin{border-strip tableau} of shape $\lambda/\mu$ and \defin{type} $\nu$
is a tiling of the Young diagram of $\lambda/\mu$
with labeled border-strips $B_1,\dotsc,B_\ell$
with the following properties:
\begin{itemize}
	\item the border-strip $B_j$ has label $j$ and size $\nu_j$,
	\item labeling all cells in $B_j$ with $j$ results in a labeling of the diagram
	$\lambda/\mu$ where labels in every row and every column are weakly increasing.
\end{itemize}
The \defin{height} of a border-strip tableau $T$, or any tiling of a tableau with border-strips, is the sum of the heights of the border-strips in the partition.
%
We let $\BST(\lambda/\mu,\nu)$ denote the set of all such border-strip tableaux.
\end{definition}
We use the convention to place the label of a strip in its tail, as done in \cref{fig:BSTExample}.
\begin{remark}
$\BST(\lambda/\mu,1^{n})$ can be identified with
the set of standard Young tableaux of shape $\lambda/\mu$.
\end{remark}

\begin{figure}[!ht]\centering
\begin{subfigure}[b]{.35\textwidth}
	\ytableausetup{boxsize=1.2em}
	\[
	\ytableaushort{\none\none{}{}{}{},\none\none{},{}{}{},{x}}
	\]
	\caption{A border-strip of height $3$. Its tail is labelled with $x$.}
	\label{fig:borderStripExample}
\end{subfigure}\quad
\begin{subfigure}[b]{.6\textwidth}\centering
	\begin{tikzpicture}[baseline=(current bounding box.center)]
		\begin{scope}[yscale=-1]
			%core
			\draw[black,x=1em,y=1em,fill=lightgray] (0,0)--(2,0)--(2,1)--(1,1)--(1,4)--(0,4)--(0,0);
			% %1
			% \begin{scope}[yshift=1em]
			% \pic{iTrio};
			% \end{scope}
			%2
			\begin{scope}[yshift=4em]
				\pic{iTrio};
			\end{scope}
			%3
			\begin{scope}[xshift=1em,yshift=0em]
				\pic{jTrio};
			\end{scope}
			%4
			\begin{scope}[xshift=3em,yshift=0em]
				\pic{tTrio};
			\end{scope}
			%5
			\begin{scope}[xshift=1em,yshift=2em]
				\pic{hTrio};
			\end{scope}
			%6
			\begin{scope}[xshift=1em,yshift=3em]
				\pic{hTrio};
			\end{scope}
			%7
			\begin{scope}[xshift=4em,yshift=1em]
				\pic{iTrio};
			\end{scope}
			%8
			\begin{scope}[xshift=5em,yshift=0em]
				\pic{iTrio};
			\end{scope}
			%9
			\begin{scope}[xshift=6em,yshift=0em]
				\pic{tTrio};
			\end{scope}
			%10
			\begin{scope}[xshift=1em,yshift=4em]
				\pic{tTrio};
			\end{scope}
			%11
			\begin{scope}[xshift=2em,yshift=4em]
				\pic{jTrio};
			\end{scope}
			%12
			\begin{scope}[xshift=4em,yshift=3em]
				\pic{jTrio};
			\end{scope}
			%
			\begin{scope}[xshift=7em,yshift=0em]
				\pic{jTrio};
			\end{scope}
			\begin{scope}[xshift=-0.5em,yshift=-0.5em]
				\node[x=1em,y=1em] at (1,4) {};
				\node[x=1em,y=1em] at (1,7) {1};
				\node[x=1em,y=1em] at (2,2) {2};
				\node[x=1em,y=1em] at (4,2) {3};
				\node[x=1em,y=1em] at (2,3) {4};
				\node[x=1em,y=1em] at (2,4) {5};
				\node[x=1em,y=1em] at (5,4) {6};
				\node[x=1em,y=1em] at (6,3) {7};
				\node[x=1em,y=1em] at (7,2) {8};
				\node[x=1em,y=1em] at (2,6) {9};
				\node[x=1em,y=1em] at (3,6) {\small{10}};
				\node[x=1em,y=1em] at (5,5) {\small{11}};
				\node[x=1em,y=1em] at (8,2) {\small{12}};
			\end{scope}
		\end{scope}
	\end{tikzpicture}
	\caption{A border-strip tableaux in $\BST((9^2,\!6^3,\!4,\!1)/(2,\!1^3), 3)$ of height $13$. The labels are placed in the tails of each strip.}\label{fig:BSTExample}
	\end{subfigure}


	\caption{
	A border-strip and a border-strip tableaux.
	}

\end{figure}

Finally, let us recall that the \defin{skew character}
$\chi^{\lambda/\mu}$ corresponding to a skew shape $\lambda/\mu$ can
be defined as
\begin{equation*}\label{eq:characterLRRule}
  \chi^{\lambda/\mu} = \sum_{\nu} c^{\lambda}_{\mu,\nu} \chi^{\nu},
\end{equation*}
where the Littlewood--Richardson coefficients $c^{\lambda}_{\mu,\nu}$
are the structure constants for the expansion
$s_{\lambda/\mu} = \sum_\nu c^{\lambda}_{\mu,\nu} s_\nu$ of a skew
Schur function into Schur functions, among many other
interpretations.

We can now state the Murnaghan--Nakayama rule.
\begin{theorem}[{Murnaghan--Nakayama, see \cite[Cor. 7.17.5 ]{StanleyEC2}}]\label{thm:mnRule}
The skew characters are given by the signed sum
\[
\chi^{\lambda/\mu}(\nu) = \sum_{B \in \BST(\lambda/\mu,\nu)} (-1)^{\height(B)}.
\]
\end{theorem}



In the remainder of the paper, we shall only concern ourselves with
border-strip tableaux where all strips have the same size~$m$, which we denote by \defin{$\BST(\lambda/\mu,m)$}.
In this case, the parity of the height depends only on the shape $\lambda/\mu$. It follows that the signed sum in the Murnaghan--Nakayama rule, \cref{thm:mnRule}, is cancellation-free. In particular, we have:

\begin{corollary}\label{cor:characterAsBST}
	Let $\lambda/\mu$ be a skew shape of size $n=dm$.
	Then
	\[
	f^{\lambda/\mu}(\xi^d) = \chi^{\lambda/\mu}((m^d)) = \varepsilon |\BST(\lambda/\mu,m)|,
	\]
	where $\xi$ is a primitive $n^\thsup$ root of unity and
	$\varepsilon=(-1)^{\height(B)}$ for any
	$B\in \BST\left(\lambda/\mu, m \right)$.
\end{corollary}
\begin{remark}
  For the non-skew case, the first equation is due to
  T.~Springer~\cite[Prop. 4.5]{Springer1974}, whereas the second was
  proved by G.~James and
  A.~Kerber~\cite[Thm. 2.7.27]{JamesKerber1984} using the abacus
  model for the quotient of a partition.
\end{remark}

In order to apply \cref{thm:alexanderssonAmini}, we need bounds on the number of border-strip tableaux.
We shall first recall the following theorem by S.~Fomin and N.~Lulov, for partitions.
\begin{theorem}[\cite{FominLulov1997}]
For any partition $\lambda \vdash n$, we have
\begin{equation}\label{eq:qndDef}
  |\BST(\lambda, d)| \leq \sqrt[d]{\frac{d^n}{\binom{n}{n/d,\dots,n/d}}} \cdot  |\BST(\lambda, 1)|^{1/d}
\end{equation}
\end{theorem}

From this we establish a new bound for the number of border-strip tableaux of skew shapes.

\begin{theorem}\label{thm:BST-bound-skew}
	Let $\lambda/\mu$ be a skew shape with $n$ cells and let $k$ be a
	positive integer with $k\mid n$.  Suppose that
	$|\BST(\lambda/\mu, k)| \geq 2$.  Then
	\[
	|\BST(\lambda/\mu, k)| \geq %
	\sum_{d\mid\frac{n}{k}, d > 1} |\BST(\lambda/\mu, dk)|.
	\]
	Additionally, the inequality holds if $n/k$ is a prime number.
\end{theorem}
\begin{remark}
  Our proof of this theorem is relatively complicated.  We use
  \cref{eq:qndDef} to prove it in the case of partitions and $k=1$.
  Then, using the Littlewood--Richardson rule, we extend this to the
  case of skew shapes and $k=1$. Finally, we deduce the general case
  using the abacus model for the quotient of a skew shape.
\end{remark}

\begin{example}
    For $\lambda=(10,\!1^2) \vdash 12$ we have \begin{align*}
        |\BST(\lambda, 3)| & = 1, & |\BST(\lambda, 6)|+|\BST(\lambda, 12)| &= 2.\end{align*}  Therefore, the
  condition that $|\BST(\lambda, k)|\geq 2$ is necessary.
\end{example}
\begin{remark}
  For $k=1$, apart from the single row and single column partitions,
  there are only three shapes $\lambda/\mu$ where equality is
  attained: $(2^2)$, $(3^2)$ and $(2^3)$.  Other than that, the
  minimal difference between the two sides of the inequality is
  attained for hooks of the form $(n-1,\!1)$.  In this case it equals
  $n-\tau(n)$, where $\tau(n)$ is the number of divisors of $n$.
\end{remark}


\section{Cyclic sieving for skew standard tableaux}\label{sec:mainCSP}

In this section we apply the bounds established in the previous
section and \cref{thm:alexanderssonAmini} to prove the existence
of several cyclic sieving phenomena for various families of skew
standard Young tableaux.


From \cref{thm:BST-bound-skew} and \cref{cor:characterAsBST} we obtain:
\begin{proposition}\label{cor:mainCSPInequality}
  Let $\lambda/\mu$ be a non-empty skew shape with $n$ cells, let
  $m\in\setNzero$ and let $k$ be a positive integer with $k \mid n$.  Then
  \[
  \sum_{d\mid k}\mu(k/d) |f^{\lambda/\mu}(\xi^d)|^m \geq 0,
  \]
  for a primitive $n^\thsup$ root of unity $\xi$.
\end{proposition}

\begin{remark}
  One might think that $|f^{\lambda/\mu}(\xi^d)|$ could be the
  number of fixed points of a group action, despite the fact that
  $|f^{\lambda/\mu}(q)|$ is not a polynomial.  However, this is not
  the case.
  For example, consider $\lambda=(2,1)$.  Then $f^\lambda(q) = q+q^2$
  and, for a $3^\rdsup$ root of unity $\xi$, we have
  $|f^\lambda(\xi^3)|=|\BST(\lambda,1)|=2$ and
  $|f^\lambda(\xi)|=|\BST(\lambda,3)|=1$.  This is incompatible with the
  possible orbit sizes of a group action on a set with two elements.
  Indeed, for $k=3$ we obtain
  \[
    \frac{1}{k}\sum_{d\mid k} \mu(k/d) |f^\lambda(\xi^d)| = \frac{1}{3}(-1 + 2),
  \]
  which, by \cref{rmk:orbit-sizes}, would have to be an integer.
\end{remark}

Taking into account the previous remark, it makes sense to look for
shapes $\lambda/\mu$ such that the character $f^{\lambda/\mu}$
evaluated at roots of unity is nonnegative.
\begin{theorem}\label{prop:positive-cyclic-sieving}
  Let $\lambda/\mu$ be a skew shape with $n$ cells and let
  $m$ be a positive integer.  Then there is a cyclic group action $\rho$ of order $n$ such that
  \[
  \left(
    \underbrace{\SYT(\lambda/\mu)\times\dots\times\SYT(\lambda/\mu)}_m,
    \langle \rho \rangle, f^{\lambda/\mu}(q)^m \right)
  \]
  exhibits the cyclic sieving phenomenon if and only if $m$ is even,
  or $m$ is odd and for each positive integer $k$ with $k\mid n$, every 
  tiling of $\lambda/\mu$ with strips of size $k$ has even height.
\end{theorem}
\begin{remark}
  The case $m=2$ of this theorem does not extend to squares of arbitrary
  representations of the symmetric group.  For example, consider the
  representation with character $\chi^{(4)} + \chi^{(2,1^2)}$.  Its
  fake degree polynomial is $f(q) = 1+q^3+q^4+q^5$.  Then we obtain,
  for a primitive fourth root of unity $\xi$, that $f(\xi)^2 = 4$ and
  $f(\xi^2)^2 = 0$.  This violates the condition in
  \cref{thm:alexanderssonAmini} for $k=2$, because
  $\mu(2)f(\xi)^2 + \mu(1)f(\xi^2)^2 = -4$.
\end{remark}

\begin{corollary}
  Let $\lambda = (a, 1^{n-a})$ be a hook-shaped partition of $n$.
  Then there is a cyclic group action $\rho$ such that
  $\left( \SYT(\lambda), \langle \rho \rangle, f^{\lambda}(q)
  \right)$ exhibits the cyclic sieving phenomenon if and only if $n$ and $a$ are odd and
  $a-1 \mod m$ is even for $m\mid n$, $1\leq m < a$.
\end{corollary}
\begin{remark}
  According to the previous theorem, for $\lambda=(3,1^{n-3})$ a cyclic
  group action of order $n$ with character $f^\lambda(q) = q^{(n-2)(n-3)/2}\frac{[n-1]_q[n-2]_q}{[2]_q}$ exists for all
  odd $n>3$.  In this case, there should be one singleton orbit and
  $(n-3)/2$ orbits of size $n$.  Indeed, an appropriate group action
  can be constructed as follows:

  Identify a tableau with the two labels $x < y$ different from $1$
  in the first row.
  Note that $y-x \in \{1,2,\dotsc,n-2\}$, and only the pair $(2,n)$
  has difference $n-2$.
  We let the generator of the group action $\eta$ act as follows:
  \[
   \eta(x,y) \coloneqq
   \begin{cases}
    (2,n) &\text{ if } x=2,\; y=n, \\
    (x+2,y+2) &\text{ if } 2\leq x < y \leq n-2,\\
    (2,x+1) &\text{ if } y=n-1, \\
    (3,x+1) &\text{ if }x>2,\; y=n. \\
   \end{cases}
  \]
  We then note that if $(u,v)=\eta(x,y)$, then
  $v-u \in \{y-x, (n-2) - (y-x) \}$.
  This explains why there are $(n-3)/2$ orbits of length $n$.
  We leave the remaining details to the reader.
\end{remark}

A different way to ensure positivity of the character
$f^{\lambda/\mu}$ is to decrease the order of the cyclic group.
\begin{theorem}\label{thm:stretchedCSPtheorem}
Let $\lambda/\mu$ be a skew shape such that every row contains a multiple of $m$ cells.
Then there is a cyclic group action $\rho$ of order $m$ such that
\[
 \left( \SYT(\lambda/\mu), \langle\rho\rangle, f^{\lambda/\mu}(q) \right)
\]
exhibits the cyclic sieving phenomenon.
\end{theorem}

\section{Permutations and invariants of the adjoint representation of
  \texorpdfstring{$\mathrm{GL}_n$}{GLn}}\label{sec:gln}
In this section we apply our results to study the space of invariants
of tensor powers of the adjoint representation $\gl_r$ of the general
linear group $\GL_r$.


\begin{definition}
  The \defin{rotation} $\rot\sigma$ of a permutation
  $\sigma\in\symS_n$ is the permutation obtained by conjugating with
  the long cycle $\longcycle$.
\end{definition}
\begin{remark}
  Equivalently, let $D_\sigma$ be the chord diagram associated
  with $\sigma$, that is, the directed graph with vertices
  $\{1,\dotsc,n\}$ arranged counterclockwise on a circle, and arcs
  $(i,\sigma(i))$.  Then $D_{\rot\sigma}$ is the chord diagram
  obtained by rotating the graph clockwise.
  See \cref{fig:rotation-permutation} for an illustration.
\end{remark}
\begin{figure}[!ht]
  \tikzset{->-/.style={decoration={
        markings,
        mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\[
  [5,\!4,\!1,\!2,\!3] = \begin{tikzpicture}[line width=1pt, baseline={([yshift=-1ex]current bounding box.center)}]
    \node (a) [draw=none, minimum size=2cm, regular polygon, regular polygon sides=5] at (0,0) {};
    \foreach \n in {1,2,...,5}
        \path (a.center) -- (a.corner \n) node[pos=1.15] {$\n$};
    \draw[->-=.9] (a.corner 1) to (a.corner 5);
    \draw[->-=.9, bend left] (a.corner 2) to (a.corner 4);
    \draw[->-=.9] (a.corner 3) to (a.corner 1);
    \draw[->-=.9, bend left] (a.corner 4) to (a.corner 2);
    \draw[->-=.9] (a.corner 5) to (a.corner 3);
  \end{tikzpicture}
  \qquad\stackrel{\rot}{\mapsto}\qquad
    \begin{tikzpicture}[line width=1pt, baseline={([yshift=-1ex]current bounding box.center)}]
    \node (a) [draw=none, minimum size=2cm, regular polygon, regular polygon sides=5] at (0,0) {};
    \foreach \n in {1,2,...,5}
        \path (a.center) -- (a.corner \n) node[pos=1.15] {$\n$};
    \draw[->-=.9] (a.corner 5) to (a.corner 4);
    \draw[->-=.9, bend left] (a.corner 1) to (a.corner 3);
    \draw[->-=.9] (a.corner 2) to (a.corner 5);
    \draw[->-=.9, bend left] (a.corner 3) to (a.corner 1);
    \draw[->-=.9] (a.corner 4) to (a.corner 2);
  \end{tikzpicture}= [3,\!5,\!1,\!2,\!4]
\]
\caption{Rotation of $\pi = [5,\!4,\!1,\!2,\!3]$ as rotation of the chord diagram. Note that $\shape([5,\!4,\!1,\!2,\!3])= (3,1^2)$ and $\shape([3,\!5,\!1,\!2,\!4])= (3,2)$.}
\label{fig:rotation-permutation}
\end{figure}



The following theorem makes the character of rotation explicit.
\begin{theorem}[\cite{BarceloReinerStanton2008,Rhoades2010b,RubeyWestbury2014}]\label{thm:rhoadesPMatrix}
  \[
    \left( \symS_n, \langle \rot \rangle,  \sum_{\lambda \vdash n}   f^{\lambda}(q)^2  \right)
  \]
  exhibits the cyclic sieving phenomenon.
\end{theorem}

\begin{definition}
Recall that the \defin{Robinson--Schensted correspondence} provides a bijection
\[
 \symS_n
 \leftrightarrow
 \{
 (P,Q) \in \SYT(\lambda)\times \SYT(\lambda) : \lambda \vdash n
 \}.
\]
The \defin{shape} $\shape(\sigma)$ of a permutation $\sigma$ is the
common shape of the standard Young tableaux $P$ and $Q$
corresponding to $\sigma$ under the Robinson--Schensted
correspondence.  We let \defin{$R_\lambda$} denote the set of permutations of shape $\lambda$.
\end{definition}

By \cref{prop:positive-cyclic-sieving} there exists an action of the cyclic group of
order $n$ on $R_\lambda$ with character $(f^\lambda(q))^2$. Taking the direct sum of the underlying linear representations over all partitions of $n$ we obtain by \cref{thm:rhoadesPMatrix} and \cref{thm:Brauer}, the following.
\begin{theorem}\label{cor:rotationInvatiantStat}
  Let $P_n$ be the set of partitions of $n$.  Then there exists a map
  $\st:\symS_n\to P_n$ which is invariant under rotation and
  equidistributed with the Robinson--Schensted shape.  That is,
  \[
    \st\circ \rot = \st\qquad\text{and}\qquad %
    \# \{ \sigma \in \symS_n : st(\sigma) = \mu \} = \# \{ \sigma \in \symS_n : sh(\sigma) = \mu \}
  \]
  for all $\mu \vdash n$. Moreover, with
  $\symS_n^\lambda\coloneqq\{\pi\in\symS_n: \st(\sigma)=\lambda\}$,
  the triple
  \[
    (\symS_n^\lambda, \langle\rot\rangle, f^\lambda(q)^2)
  \]
  exhibits the cyclic sieving phenomenon.
\end{theorem}
\begin{remark}
  We stress that we are currently unable to present such a statistic
  explicitly.
\end{remark}

We now turn to the connection with the invariants of tensor powers of
the adjoint representation of $\GL_r$, which is the original
motivation for this article.

Let $V$ be an $r$-dimensional complex vector space and let
$\gl_r = \End(V)$ be the adjoint representation
$\GL_r\to\End(\gl_r)$, $A\mapsto T A T^{-1}$.  We consider the action
of the symmetric group $\symS_n$ on $\gl_r^{\otimes n}$ which permutes tensor positions.
This action is inherited by the subspace
$\left(\gl_r^{\otimes n}\right)^{\GL_r}$ of $\GL_r$-invariants.
Using Schur--Weyl duality we obtain the following.
\begin{lemma}[\cite{RubeyWestbury2014}]\label{thm:invariants-character}
  Let $\gl_r$ be the adjoint representation of $\GL_r$, and, given a
  partition $\lambda\vdash n$, let $S_\lambda$ be the corresponding
  irreducible representation of the symmetric group.  Then there is
  an isomorphism
  \[
  \left(\gl_r^{\otimes n}\right)^{\GL_r}\cong%
  \bigoplus_{\substack{\lambda\vdash n\\\ell(\lambda)\leq r}}
  S_\lambda\otimes S_\lambda.
  \]
\end{lemma}
\begin{theorem}\label{thm:existence-pr-rot}
  There is a basis of
  $\left(\gl_r^{\otimes n}\right)^{\GL_r}$ which is preserved by the
  action of the long cycle.  Moreover, this action is isomorphic to
  the action of rotation on
  \[
    \symS_n^{(r)}\coloneqq\{\pi\in\symS_n: \ell(\st(\pi)) \leq
    r\} = \bigcup\limits_{\substack{\lambda\vdash n\\\ell(\lambda)\leq r}}\symS_n^{\lambda}.
  \]
\end{theorem}
\begin{proof}
  \cref{cor:rotationInvatiantStat} together with
  \cref{thm:invariants-character} shows that the linear
  representation corresponding to the action of rotation on
  $\symS_n^{(r)}$ and
  \[
    \bigoplus_{\substack{\lambda\vdash n\\\ell(\lambda)\leq r}}
    S_\lambda\otimes S_\lambda\tocyclic
  \]
  have the same character.
\end{proof}

\begin{remark}
  There is a notion of promotion for so called \defin{alternating
    tableaux}, that is, highest weight words in the crystal for
  $\gl_r^{\otimes n}$.  It follows from our results and a theorem of
  Westbury, that this action is isomorphic to the action of rotation
  on $\symS_n^{(r)}$.  For $r\geq n$ an appropriate bijection is
  given in \cite{PfannererRubeyWestbury2020}.
\end{remark}



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