Crystals, semistandard tableaux and cyclic sieving phenomenon

TL;DR

This paper establishes a new cyclic sieving phenomenon on semistandard Young tableaux linked to crystal structures, proving specific cases where cyclic actions and promotion generate a bicyclic sieving phenomenon, especially for hook shapes.

Contribution

It introduces a novel cyclic sieving phenomenon on semistandard Young tableaux derived from crystal structures and explores its connection with promotion, including a bicyclic sieving result for hook shapes.

Findings

Proves cyclic sieving for certain Young tableaux with gcd condition.

Connects cyclic action with promotion in tableaux.

Demonstrates bicyclic sieving for hook shapes.

Abstract

In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(\lambda)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $\lambda$ is a Young diagram with $\ell(\lambda) < n$ and $\gcd( n, |\lambda| )=1$, then the triple $\left( \mathsf{SST}_n(\lambda), \mathsf{C}, q^{- \kappa(\lambda)} s_\lambda(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by $\mathsf{c}$ and $\mathsf{pr}^n$ for hook shape.

Equations

$$\#X^c=f({\omega }_{o(c)}),$$

$$f(q)\equiv \sum _{l=0}^{n-1}{a}_l{q}^l(\mathrm{mod}{q}^n-1),$$

$$\left({\mathsf{SST}}_n(\lambda ),⟨\mathsf{pr}⟩,q^{-\kappa (\lambda )}{s}_{\lambda }(1,q,\dots ,q^{n-1})\right)$$

$$\left({\mathsf{SST}}_n(\lambda ,\alpha ),⟨{\mathsf{pr}}^l⟩,q^{\frac{1}{2}(a^{2}b-({\alpha }_1^2+{\alpha }_2^2+\cdots +{\alpha }_n^2))}{K}_{\lambda ,\alpha }(q)\right)$$

$$\left({\mathsf{SST}}_n(\lambda ),\mathsf{C},q^{-\kappa (\lambda )}{s}_{\lambda }(1,q,\dots ,q^{n-1})\right)$$

$$S_{\lambda }(q,t)=q^{-\kappa (\lambda )}\sum _{\mu ⊢N}{t}^{A_{\mu }}{K}_{\lambda ,\mu }(t^{\frac{\mathsf{d}}{{\mathsf{d}}_{\mu }}})\cdot m_{\mu }(1,q,q^2,\dots ,q^{n-1})$$

$$({\alpha }_i,{\alpha }_j)={\mathsf{d}}_i{a}_{ij}(i,j\in I),\text{and }⟨h_i,\lambda ⟩=\frac{2({\alpha }_i,\lambda )}{({\alpha }_i,{\alpha }_i)}(\lambda \in {\mathfrak{h}}^{\ast },i\in I).$$

$$s_i(\lambda )=\lambda -⟨h_i,\lambda ⟩{\alpha }_i\text{ for i∈I and }\lambda \in \mathsf{P}.$$

$$\mathrm{wt}(B):=\{\xi \in \mathsf{P}\mid B_{\xi }\ne \varnothing \}.$$

$$\displaystyle\mathsf{s}_{i}(b)=\left\{\begin{array}[]{ll}\tilde{f}_{i}^{\langle h_{i},\mathrm{wt}(b)\rangle}b&\text{ if }\langle h_{i},\mathrm{wt}(b)\rangle\geq 0,\\ \tilde{e}_{i}^{-\langle h_{i},\mathrm{wt}(b)\rangle}b&\text{ if }\langle h_{i},\mathrm{wt}(b)\rangle<0.\end{array}\right.$$

$$\mathrm{wt}({\mathsf{s}}_i(b))=s_i(\mathrm{wt}(b))\text{ for i∈I and b∈B(Λ).}$$

$$\mathrm{ch}B(\Lambda ):=\sum _{\xi \in \mathrm{wt}(B(\Lambda ))}|B(\Lambda {)}_{\xi }|e^{\xi },$$

$${\dim }_{q}B(\Lambda )=\sum _{\xi \in \mathrm{wt}(B(\Lambda ))}|B(\Lambda {)}_{\xi }|q^{\mathsf{ev}(\Lambda -\xi )},$$

$$\mathsf{ev}(\beta ):=\sum _{i\in I}{b}_i\text{ for }\beta =\sum _{i\in I}{b}_i{\alpha }_i\in \mathsf{Q}.$$

$$\mathsf{c}:={\mathsf{s}}_1{\mathsf{s}}_2\cdots {\mathsf{s}}_r.$$

$$1$$

$$s_{\lambda }(x_1,\dots ,x_n):=\sum _{T\in {\mathsf{SST}}_n(\lambda )}{x}^T.$$

$$\textstyle{T={\small\begin{tabular}[]{|c|c|c|}\hline\cr 1&1&2\\ \hline\cr 3&3&4\\ \hline\cr 4\\ \cline{1-1}\cr\end{tabular}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$

$${\alpha }_i:={ϵ}_i-{ϵ}_{i+1}\text{ and }{\varpi }_i:=\sum _{k=1}^i{ϵ}_i.$$

$${\mathsf{SST}}_n(\lambda )\simeq B(\mathrm{wt}(\lambda ))$$

$$s_{\lambda }(1,q,\dots ,q^{n-1})=q^{\kappa (\lambda )}{\dim }_{q}B(\mathrm{wt}(\lambda )),\text{where κ(λ)=∑k=1ℓ(k−1)λk.}$$

$$\mathrm{wt}(\mathsf{c}(T))=\mathrm{wt}(\mathsf{pr}(T))=c(\mathrm{wt}(T))\text{ for T∈SSTn(λ).}$$

$$\mathsf{ev}(c(\beta ))\equiv \mathsf{ev}(\beta )(\mathrm{mod}n).$$

$$\mathsf{ev}(c(\Lambda ))\equiv \mathsf{ev}(\Lambda )-N(\mathrm{mod}n).$$

$$c(\alpha_{i})=\left\{\begin{array}[]{ll}\alpha_{i+1}&\text{ if }i\neq n-1,\\ -\alpha_{1}-\alpha_{2}-\cdots-\alpha_{n-1}&\text{ if }i=n-1.\end{array}\right.$$

$$\mathsf{ev}(c({\varpi }_i))=\mathsf{ev}({\varpi }_i)-i,$$

$$c(\Lambda )-\Lambda \in \mathsf{Q}\text{ and }{c}^k(\beta )\in \mathsf{Q}\text{ for }k\in {\mathbb{Z}}_{\ge 0},$$

$$\mathsf{ev}(c^k(c-\mathrm{id})\beta )$$

$$\mathsf{ev}(c^k(c-\mathrm{id})\Lambda )$$

$$\mathsf{ev}(c^k(c-\mathrm{id})\xi )=\mathsf{ev}(c^k(c-\mathrm{id})(\Lambda -\beta ))\equiv -N(\mathrm{mod}n),$$

$$\mathsf{ev}(c^t(\xi ))-\mathsf{ev}(\xi )\equiv \mathsf{ev}((c^t-\mathrm{id})(\xi ))\equiv \sum _{k=0}^{t-1}\mathsf{ev}(c^k(c-\mathrm{id})\xi )\equiv -t\cdot N(\mathrm{mod}n).$$

Full text

The research of Y.-T. Oh was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2018R1D1A1B07051048).

The research of E. Park was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2017R1A1A1A05001058).

Introduction

The cyclic sieving phenomenon was introduced in 2004 by Reiner-Stanton-White in [14]. Let $X$ be a finite set, with an action of a cyclic group $C$ of order $n$, and $f(q)$ a polynomial in $q$ with nonnegative integer coefficients. For $d\in\mathbb{Z}_{>0}$, let $\omega_{d}$ be a $d$th primitive root of the unity. We say that $(X,C,f(q))$ exhibits the cyclic sieving phenomenon if, for all $c\in C$, we have

[TABLE]

where $o(c)$ is the order of $c$ and $X^{c}$ is the fixed point set under the action of $c$. Note that this condition is equivalent to the following:

[TABLE]

where $a_{l}$ counts the number of $C$-orbits on $X$ for which the stablilizer-order divides $l$. It has since then been extensively investigated for various combinatorial objects with an action of a finite cyclic group including words, multisets, permutations, non-crossing partitions, lattice paths, tableaux (see [16] for details).

In [17, 18], Schuzenberger introduced the promotion operator $\mathsf{pr}$ on (semi)standard Young tableaux, which takes one (semi)standard Young tableau to another via jeu de taquin slides. Afterwards, it has been studied widely and now has become one of the important objects in various research areas (see [19]). It is known that it has a finite order, but in the best knowledge of the authors, its order is still mysterious except a few cases such as rectangular or staircase Young diagrams [5, 13].

Given a Young diagram $\lambda$, let $\mathsf{SST}_{n}(\lambda)$ be the set of semistandard Young tableaux of shape $\lambda$ with entries in $\{1,2,\ldots,n\}$. In [15], Rhoades proved representation-theoretically that if $\lambda$ is of rectangular shape, the triple

[TABLE]

exhibits the cyclic sieving phenomenon, where $\kappa$ is the statistic on $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ given by $\kappa(\lambda)=\sum_{i\geq 1}(i-1)\lambda_{i}$, and $s_{\lambda}(1,q,\ldots,q^{n-1})$ is the principal specialization of the Schur polynomial $s_{\lambda}(x_{1},x_{2},\ldots,x_{n})$. This result, however, is no longer valid outside rectangular shape in general, which says that, for a non-rectangular shape, another appropriate operator other than $\mathsf{pr}$ should be considered if we stick to the principal specialization on $\mathsf{SST}_{n}(\lambda)$. In [4], Rhoades’ result is refined in the following manner. Let $\lambda=(a^{b})$ and $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ be a composition of $ab$, such that $\alpha$ is invariant under $l$th cyclic shift, then the triple

[TABLE]

exhibits the cyclic sieving phenomenon, where $K_{\lambda,\alpha}(q)$ is a Kostka-Foulkes polynomial associated with $\lambda$ and $\alpha$. Unfortunately, outside rectangular case, no results similar to this seem to be known yet.

In the present paper, we investigate the cyclic sieving phenomenon on $\mathsf{SST}_{n}(\lambda)$ with a cyclic action arising from its crystal structure (see Section 1 for crystals). For this purpose, we first notice that $\mathsf{pr}=\sigma_{1}\sigma_{2}\cdots\sigma_{n-1}$, where $\sigma_{i}$ is the $i$th Bender–Knuth involution acting on $\mathsf{SST}_{n}(\lambda)$. In general, $\sigma_{i}$’s do not satisfy braid relations. We then note that $\mathsf{SST}_{n}(\lambda)$ has a $U_{q}(\mathfrak{sl}_{n})$-crystal structure, thus it is equipped with an action of the Weyl group. Hence it would be very natural to consider the operator $\mathsf{c}:=\mathsf{s}_{1}\mathsf{s}_{2}\cdots\mathsf{s}_{n-1}$ on $\mathsf{SST}_{n}(\lambda)$, where $\mathsf{s}_{i}$ are simple reflections in the Weyl group. The operator $\mathsf{c}$ shares several similarities with $\mathsf{pr}$, for instance, it is easy to check that $\mathrm{wt}(\mathsf{c}(T))=\mathrm{wt}(\mathsf{pr}(T))=s_{1}s_{2}\cdots s_{n-1}(\mathrm{wt}(T))$. One of the most favorable features of $\mathsf{c}$, compared with $\mathsf{pr}$, might be that its order is given by $n$ for arbitrary shape $\lambda,$ whereas the order of $\mathsf{pr}$ is very difficult to compute.

In the viewpoint of crystal theory, by using the operator $\mathsf{c}$ instead of $\mathsf{pr}$, we observe a new cyclic sieving phenomenon on $\mathsf{SST}_{n}(\lambda)$ beyond rectangular shape. More precisely, we prove that if $\lambda$ is a Young diagram with $\ell(\lambda)<n$ and $\gcd(n,|\lambda|)=1$, then the triple

[TABLE]

exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$ (see Theorem 3.3). There are several examples for which our cyclic sieving phenomenon hold without the condition $\gcd(n,|\lambda|)=1$, and Remark 3.4 shows an example for another cyclic sieving phenomenon with a specialization of $s_{\lambda}$ other than the principal specialization. It would be an interesting problem to give a characterization of Young diagrams $\lambda$ such that $\left(\mathsf{SST}_{n}(\lambda),c,f(q)\right)$ exhibits a cyclic sieving phenomenon, where $f(q)$ is a suitable specialization of $s_{\lambda}$ (multiplied by a $q$-power). We also remark that the cyclic sieving phenomenon on the set of isolated vertices of a tensor product $B^{\otimes m}$ of a crystal $B$ with a different cyclic operator was studied in [20].

Next, we turn to the connection between $\mathsf{c}$ and the $\mathsf{pr}$. For an $n$-tuple $\alpha\in\mathbb{Z}_{\geq 0}^{n}$, let $\mathsf{SST}_{n}(\lambda,\alpha):=\{T\in\mathsf{SST}_{n}(\lambda)\mid\mathrm{cont}(T)=\alpha\}$. We denote by $\mathrm{cont}(\lambda)$ the set of all contents of $T$ where $T$ varies over $\mathsf{SST}_{n}(\lambda)$, and by $\mathrm{cont}^{+}(\lambda)$ the set of all $\alpha=(a_{1},\ldots,a_{n})\in\mathrm{cont}(\lambda)$ such that $a_{1}\geq a_{2}\geq\cdots\geq a_{n}$. Notice that $\mathsf{SST}_{n}(\lambda,\alpha)$ is invariant under $\mathsf{pr}^{n}$ for any $\alpha\in\mathrm{cont}(\lambda)$. For clarity, denote by $\mathsf{pr}^{n}|_{\alpha}$ the restriction of $\mathsf{pr}^{n}$ to $\mathsf{SST}_{n}(\lambda,\alpha)$.

We here deal with the case where $\lambda$ is of hook shape or two-column shape. In these special cases, we show that $\mathsf{pr}^{n}$ commutes with $\mathsf{s}_{i}$’s, thus $\mathsf{pr}^{n}$ commutes with $\mathsf{c}$. We then show that the order of $\mathsf{pr}^{n}$ on $\mathsf{SST}_{n}(\lambda)$ equals $\mathrm{lcm}\{\mathfrak{o}_{\lambda}(\alpha)\mid\alpha\in\mathrm{cont}^{+}(\lambda)\}$, where $\mathfrak{o}_{\lambda}(\alpha)$ denotes the order of $\mathsf{pr}^{n}|_{\alpha}$ and $\mathrm{lcm}\{k_{1},k_{2},\ldots,k_{t}\}$ the least common multiple of $k_{1},k_{2},\ldots,k_{t}$. We next consider the bicyclic sieving phenomenon on $\mathsf{SST}_{n}(\lambda)$ in case where $\lambda$ is of hook shape with $(n,|\lambda|)=1$ (see [16, Section 9] for the definition). Let $\lambda=(N-m,1^{m})$ with $\gcd(n,N)=1$, and consider the polynomial

[TABLE]

given in Theorem 4.10. Here $m_{\mu}(x_{1},x_{2},\ldots,x_{n})$ is the monomial symmetric polynomial assocoated to $\mu$, and $K_{\lambda,\mu}(t)$ is the Kostka-Foulkes polynomial associated with $\lambda$ and $\mu$. Note that the evaluation $S_{\lambda}(q,t)$ at $t=1$ is equal to $q^{-\kappa(\lambda)}s_{\lambda}(1,q,\ldots,q^{n-1})$. We show that the triple $(\mathsf{SST}_{n}(\lambda),\mathsf{C}\times\mathsf{P},S_{\lambda}(q,t))$ exhibits the bicyclic sieving phenomenon, where $\mathsf{P}$ is the cyclic group generated by $\mathsf{pr}^{n}$ (see Theorem 4.10).

This paper is organized as follows: In Section 1, we review briefly the crystal theory. In Section 2, we recall the combinatorics of Young tableaux. In Section 3, we study the action of $\mathsf{c}$ on $\mathsf{SST}_{n}(\lambda)$ and prove the triple $\left(\mathsf{SST}_{n}(\lambda),\mathsf{C},q^{-\kappa(\lambda)}s_{\lambda}(1,q,\ldots,q^{n-1})\right)$ exhibits the cyclic sieving phenomenon. In Section 4, we investigate a connection between $\mathsf{c}$ and $\mathsf{pr}$ and show the bicyclic sieving phenomenon given by $\mathsf{c}$ and $\mathsf{pr}^{n}$ for hook shape.

1. Crystals

Let $I$ be a finite index set. A square matrix $\mathsf{A}=(a_{ij})_{i,j\in I}$ is called a generalized Cartan matrix if it satisfies (i) $a_{ii}=2$ for $i\in I$ and $a_{ij}\in\mathbb{Z}_{\leq 0}$ for $i\neq j$, (ii) $a_{ij}=0$ if and only if $a_{ji}=0$, (iii) there exists a diagonal matrix $D={\rm diag}(\mathsf{d}_{i}\mid i\in I)$ such that $D\mathsf{A}$ is symmetric. A Cartan datum $(\mathsf{A},\mathsf{P},\Pi,\mathsf{P}^{\vee},\Pi^{\vee})$ consists of

1. (1)

a generalized Cartan matrix $\mathsf{A}$, 2. (2)

a free abelian group $\mathsf{P}$, called the weight lattice, 3. (3)

$\Pi=\{\alpha_{i}\mid i\in I\}\subset\mathsf{P}$, called the set of simple roots, 4. (4)

$\mathsf{P}^{\vee}=\mathrm{Hom}_{\mathbb{Z}}(\mathsf{P},\mathbb{Z})$, called the coweight lattice, 5. (5)

$\Pi^{\vee}=\{h_{i}\in\mathsf{P}^{\vee}\mid i\in I\}$, called the set of simple coroots,

which satisfy

1. (1)

$\langle h_{i},\alpha_{j}\rangle=a_{ij}$ for $i,j\in I$, 2. (2)

$\Pi$ is linearly independent over $\mathbb{Q}$, 3. (3)

for each $i\in I$, there exists $\varpi_{i}\in\mathsf{P}$, called the fundamental weight, such that $\langle h_{j},\varpi_{i}\rangle=\delta_{j,i}$ for all $j\in I$.

We set $\mathsf{Q}:=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}$, called the root lattice, and $\mathsf{Q}^{+}:=\sum_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}$. We fix a nondegenerate symmetric bilinear form $(\cdot\,,\cdot)$ on $\mathfrak{h}^{*}:=\mathbb{Q}\otimes_{\mathbb{Z}}\mathsf{P}$ satisfying

[TABLE]

Let us denote by $\mathsf{P}^{+}:=\{\lambda\in\mathsf{P}\mid\langle h_{i},\lambda\rangle\geq 0\ \text{for all }\ i\in I\}$ the set of dominant integral weights, and define ${\rm ht}(\beta):=\sum_{i\in I}k_{i}$ for $\beta=\sum_{i\in I}k_{i}\alpha_{i}\in\mathsf{Q}^{+}$. Let $\mathsf{W}$ be the Weyl group associated with $\mathsf{A}$, which is generated by

[TABLE]

Let $U_{q}(\mathfrak{g})$ be the quantum group associated with the Cartan datum $(\mathsf{A},\mathsf{P},\mathsf{P}^{\vee}\Pi,\Pi^{\vee})$, which is generated by $f_{i}$, $e_{i}$ $(i\in I)$ and $q^{h}$ $(h\in\mathsf{P})$ with certain defining relations (see [6, Chater 3] for details). The notion of crystals was introduced in [7, 8, 9]. We refer the reader to [3, 6] for details.

Definition 1.1**.**

A crystal associated with $(\mathsf{A},\mathsf{P},\mathsf{P}^{\vee}\Pi,\Pi^{\vee})$ is a set $B$ together with the maps $\mathrm{wt}:B\rightarrow\mathsf{P}$, $\tilde{e}_{i},\tilde{f}_{i}:B\rightarrow B\cup\{0\}$, and $\varepsilon_{i},\varphi_{i}:B\rightarrow\mathbb{Z}\cup\{-\infty\}$ ($i\in I$) satisfying the following properties:

1. (1)

$\varphi_{i}(b)=\varepsilon_{i}(b)+\langle h_{i},\mathrm{wt}(b)\rangle$ for all $i\in I$, 2. (2)

$\mathrm{wt}(\tilde{e}_{i}b)=\mathrm{wt}(b)+\alpha_{i}$ if $\tilde{e}_{i}b\in B$, 3. (3)

$\mathrm{wt}(\tilde{f}_{i}b)=\mathrm{wt}(b)-\alpha_{i}$ if $\tilde{f}_{i}b\in B$, 4. (4)

$\varepsilon_{i}(\tilde{e}_{i}b)=\varepsilon_{i}(b)-1$, $\varphi_{i}(\tilde{e}_{i}b)=\varphi_{i}(b)+1$ if $\tilde{e}_{i}b\in B$, 5. (5)

$\varepsilon_{i}(\tilde{f}_{i}b)=\varepsilon_{i}(b)+1$, $\varphi_{i}(\tilde{f}_{i}b)=\varphi_{i}(b)-1$ if $\tilde{f}_{i}b\in B$, 6. (6)

$\tilde{f}_{i}b=b^{\prime}$ if and only if $b=\tilde{e}_{i}b^{\prime}$ for $b,b^{\prime}\in B$ and $i\in I$, 7. (7)

if $\varphi_{i}(b)=-\infty$ for $b\in B$, then $\tilde{e}_{i}b=\tilde{f}_{i}=0$.

For a crystal $B$, we set $B_{\xi}:=\{b\in B\mid\mathrm{wt}(b)=\xi\}$ so that $B=\sqcup_{\xi\in\mathsf{P}}B_{\xi}$. Let

[TABLE]

For a dominant integral weight $\Lambda\in\mathsf{P}^{+}$, we denote by $B(\Lambda)$ the crystal of the irreducible highest weight $U_{q}(\mathfrak{g})$-module $V_{q}(\Lambda)$ with highest weight $\Lambda$. For $i\in I$, we define the bijection $\mathsf{s}_{i}$ on $B(\Lambda)$ by

[TABLE]

Then the Weyl group $\mathsf{W}$ acts on the crystal $B(\Lambda)$ in which the simple reflection $s_{i}$ acts via $\mathsf{s}_{i}$ for $i\in I$ (see [3, Chapter 2.5] for details). Note that

[TABLE]

The character $\mathrm{ch}B(\Lambda)$ of $B(\Lambda)$ is defined by

[TABLE]

where $|B(\Lambda)_{\xi}|$ is the number of elements of $B(\Lambda)_{\xi}$, and $e^{\xi}$ are formal basis elements of the group algebra $\mathbb{Q}[\mathsf{P}]$ with the multiplication given by $e^{\xi}e^{\xi^{\prime}}=e^{\xi+\xi^{\prime}}$. The $q$-dimension of $B(\Lambda)$ is given by

[TABLE]

where $\mathsf{ev}:\mathsf{Q}\to\mathbb{Z}$ is the map defined as follows:

[TABLE]

We now assume that $I=\{1,2,\ldots,r\}$ and the Cartan matrix $\mathsf{A}$ is of finite type. Note that the crystal $B(\Lambda)$ is a finite set. We define the bijection $\mathsf{c}$ on $B(\Lambda)$ as follows:

[TABLE]

Since $\mathsf{s}_{i}$’s act on $B(\Lambda)$ as simple reflections of the Weyl group $\mathsf{W}$, $\mathsf{c}$ can be viewed as a Coxeter element of $\mathsf{W}$. Let $\mathsf{C}:=\langle\mathsf{c}\rangle$ be the cyclic subgroup of $\mathsf{W}$ generated by $\mathsf{c}$, and $h$ the Coxeter number of $\mathsf{W}$.

Lemma 1.2**.**

The cyclic group $\mathsf{C}$ has order $h$ and acts on the crystal $B(\Lambda)$.

2. Semistandard tableaux

For a partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{l}>0)$, the length $\ell(\lambda)$ of $\lambda$ is defined to be the number of positive parts of $\lambda$ and the size $|\lambda|$ of $\lambda$ the sum of all parts, that is, $\ell(\lambda)=l$ and $|\lambda|=\Sigma\lambda_{i}$. Throughout this paper, we will confuse $\lambda$ with its Young diagram drawn in English convention, more precisely, an array of boxes in which the $i$th row has $\lambda_{i}$ boxes from top to bottom. The conjugate $\lambda^{\prime}$ of $\lambda$ denotes the Young diagram obtained from $\lambda$ by flipping the diagonal.

A semistandard tableau $T$ of shape $\lambda$ with entries bounded by $n$ is a filling of boxes of $\lambda$ with entries in $\{1,2,\ldots,n\}$ such that

1. (1)

the entries in each row are weakly increasing from left to right, and 2. (2)

the entries in each column are strictly increasing from top to bottom.

Let $\mathsf{sh}(T)$ denote the shape of a semistandard tableau $T$ and $\mathsf{SST}_{n}(\lambda)$ the set of all semistandard tableaux of shape $\lambda$ with entries bounded by $n$. We say that $\mathbf{b}=(p,q)\in T$ if $\mathbf{b}$ is a box of $T$ at the $p$th row and the $q$th column, and denote by $T(\mathbf{b})$ the entry of the box $\mathbf{b}$. For example, the following is a semistandard tableau of shape $\lambda=(8,5,2)$ with entries bounded by $5$:

[TABLE]

For $T\in\mathsf{SST}_{n}(\lambda)$, the content $\mathrm{cont}(T)$ of $T$ is defined to be the $n$-tuple $(c_{1},\ldots,c_{n})$, where $c_{k}$ is the number of occurrences of $k$ in $T$. Setting $x^{T}:=x_{1}^{c_{1}}\cdots x_{n}^{c_{n}}$, we define the Schur polynomial

[TABLE]

Next, we describe the promotion operator $\mathsf{pr}$ on $\mathsf{SST}_{n}(\lambda)$. Let $T\in\mathsf{SST}_{n}(\lambda)$. If $T$ does not contain entries equal to $n$, then $\mathsf{pr}(T)$ is defined to be the tableau obtained from $T$ by increasing all the entries by 1. Otherwise, replace every entry equal to $n$ with a dot, then by using jeu-de-taquin, slide the dots to the northwest corner from left to right and top to bottom. Finally, replace all dots by 1’s and increase all other entries by 1 to obtain $\mathsf{pr}(T)$.

Example 2.1**.**

Let $n=4$ and $\lambda=(3,3,1)$. The following is an illustration of the promotion on a tableaux $T\in\mathsf{SST}_{4}(\lambda)$.

[TABLE]

From now on, we assume that the Cartan matrix $\mathsf{A}$ is of type $A_{n-1}$, i.e., $U_{q}(\mathfrak{g})=U_{q}(\mathfrak{sl}_{n})$, with $I=\{1,2,\ldots,n-1\}$. For $k=1,\ldots,n$, we set $\epsilon_{k}:=(0,\ldots,1,\ldots,0)\in\mathbb{Q}^{n}$ to be the unit vector with the 1 in the $k$th position. For $i\in I$, we set

[TABLE]

Then we identify the weight lattice $\mathsf{P}$ with the $n-1$-dimensional subspace of $\mathbb{Q}^{n}$ orthogonal to the vector $\epsilon_{1}+\cdots+\epsilon_{n}$. Note that the bilinear form $(\cdot\,,\cdot)$ corresponds to the usual inner product and $s_{i}(\epsilon_{j})=\epsilon_{s_{i}(j)}$ for $i\in I$, where the subscript $s_{i}$ denotes the simple transposition $(i,i+1)$ in the symmetric group $\mathfrak{S}_{n}$.

Let $\lambda=(\lambda_{1}\geq\cdots\geq\lambda_{\ell}>0)$ be a Young diagram with $\ell(\lambda)<n$. Letting $\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\ldots,\lambda_{t}^{\prime})$, we set $\mathrm{wt}(\lambda):=\sum_{k=1}^{t}\varpi_{\lambda_{k}^{\prime}}\in\mathsf{P}^{+}.$ It is well-known that $\mathsf{SST}_{n}(\lambda)$ admits a $U_{q}(\mathfrak{sl}_{n})$-crystal structure and

[TABLE]

as a $U_{q}(\mathfrak{sl}_{n})$-crystal. We refer the reader to [3, Chapter 3] and [6, Chapter 7] for details. Note that $\mathrm{wt}(T)=c_{1}\epsilon_{1}+\cdots+c_{n}\epsilon_{n}$ for $T\in\mathsf{SST}_{n}(\lambda)$, where $\mathrm{cont}(T)=(c_{1},\ldots,c_{n})$. We remark that the principal specialization of $s_{\lambda}(x_{1},\ldots,x_{n})$ is equal to the $q$-dimension of $B(\mathrm{wt}(\lambda))$ up to a power of $q$, more precisely,

[TABLE]

Since $\mathsf{SST}_{n}(\lambda)$ is a $U_{q}(\mathfrak{sl}_{n})$-crystal, the operator $\mathsf{c}$ defined as in $\eqref{Eq: def of c}$ acts on $\mathsf{SST}_{n}(\lambda)$. The lemma below follows from Lemma 1.2 immediately.

Lemma 2.2**.**

The cyclic group $\mathsf{C}$ has order $n$ and acts on the $U_{q}(\mathfrak{sl}_{n})$-crystal $\mathsf{SST}_{n}(\lambda)$.

3. Cyclic sieving phenomenon

As before, assume that the Cartan matrix $\mathsf{A}$ is of type $A_{n-1}$. Let $c:=s_{1}s_{2}\cdots s_{n-1}\in\mathfrak{S}_{n}$. Note that $\mathfrak{S}_{n}$ acts on the weight lattice $\mathsf{P}$. In addition, from the definition of $\mathsf{pr}$ and $\mathsf{c}$ it follows that

[TABLE]

Lemma 3.1**.**

1. (1)

For $\beta\in\mathsf{Q}$, we have

[TABLE] 2. (2)

Let $\Lambda\in\mathsf{P}^{+}$ and $N=\left(\Lambda,\alpha_{1}+2\alpha_{2}+\cdots+(n-1)\alpha_{n-1}\right)$. Then

[TABLE]

Proof.

(1) As $\mathsf{ev}$ is linear, it suffices to consider the case where $\beta=\alpha_{i}$ for $i\in I$. By a direct computation, we can derive that

[TABLE]

This tells us that $\mathsf{ev}(c(\alpha_{i}))\equiv\mathsf{ev}(\alpha_{i})\pmod{n}$.

(2) As above, due to the linearity of $\mathsf{ev}$, we may assume that $\Lambda=\varpi_{i}$ for $i\in I$. Note that $i=\left(\varpi_{i},\alpha_{1}+2\alpha_{2}+\cdots+(n-1)\alpha_{n-1}\right)$. It follows from the identity $c\varpi_{i}=\varpi_{i}-\alpha_{i}-\alpha_{i-1}-\cdots-\alpha_{1}$ that

[TABLE]

which justifies the assertion. ∎

For positive integers $a,b\in\mathbb{Z}_{>0}$, we denote by $\gcd(a,b)$ the greatest common divisor of $a$ and $b$. A subset $\{a_{1},a_{2},\ldots,a_{n}\}\subset\mathbb{Z}$ is called a complete residue system modulo $n$ if it has no two elements that are congruent modulo $n$.

Lemma 3.2**.**

Let $\Lambda\in\mathsf{P}^{+}$ and $N=\left(\Lambda,\alpha_{1}+2\alpha_{2}+\cdots+(n-1)\alpha_{n-1}\right)$. Suppose that $\gcd(n,N)=1$. Then, for any $\xi\in\mathrm{wt}(B(\Lambda))$, the set $\{\mathsf{ev}(\Lambda-\xi),\mathsf{ev}(\Lambda-c(\xi)),\ldots,\mathsf{ev}(\Lambda-c^{n-1}(\xi))\}$ is a complete residue system modulo $n$.

Proof.

Let $\xi\in\mathrm{wt}(B(\Lambda))$. Then we can write as $\xi=\Lambda-\beta$ for some $\beta\in\mathsf{Q}^{+}$. Since

[TABLE]

Lemma 3.1 implies that

[TABLE]

Combining these congruences, we derive that

[TABLE]

and thus, for $t=1,\ldots,n-1$,

[TABLE]

Now, our assertion follows from the assumption $\gcd(n,N)=1$. ∎

For two polynomials $f(q)$ and $g(q)$, we write $f(q)\equiv_{n}g(q)$ if $f(q)-g(q)$ is divisible by $q^{n}-1$. We are now ready to state the main result on the cyclic sieving phenomenon for semistandard tableaux.

Theorem 3.3**.**

Assume that $\lambda$ is a Young diagram with $\ell(\lambda)<n$ and $\gcd(n,|\lambda|)=1$. Then we have

1. (1)

every orbit of $\mathsf{SST}_{n}(\lambda)$ under the action of $\mathsf{C}$ is free, and 2. (2)

the triple $(\mathsf{SST}_{n}(\lambda),\mathsf{C},q^{-\kappa(\lambda)}s_{\lambda}(1,q,\ldots,q^{n-1}))$ exhibits the cyclic sieving phenomenon.

Proof.

(1) Let $\Lambda=\mathrm{wt}(\lambda)$ and denote by $\mathcal{O}(\lambda)$ the set of all orbits of $\mathsf{SST}_{n}(\lambda)$ under the action of $\mathsf{C}$. Set

[TABLE]

for each orbit $\mathcal{O}\in\mathcal{O}(\lambda)$. Also, for $T\in\mathsf{SST}_{n}(\lambda)$, we set $\mathcal{O}(T):=\{\mathsf{c}^{k}(T)\mid k\in\mathbb{Z}_{\geq 0}\}\in\mathcal{O}(\lambda).$ Since the cyclic group $\mathsf{C}$ has order $n$, we can deduce that

1. (i)

$|\mathcal{O}(T)|$ divides $n$, and 2. (ii)

$|\mathfrak{e}(\mathcal{O}(T))|\leq|\mathcal{O}(T)|$.

But, since $n\leq|\mathfrak{e}(\mathcal{O}(T))|$ due to Lemma 3.2, we can deduce that

[TABLE]

as reqired.

(2) Note that

[TABLE]

For $\mathcal{O}\in\mathcal{O}(\lambda)$, we define

[TABLE]

As $\gcd(n,|\lambda|)=1$, Lemma 3.2 implies that

[TABLE]

for any orbit $\mathcal{O}\in\mathcal{O}(\lambda).$ Combining this with the identity $|\mathcal{O}(\lambda)|=\displaystyle\frac{|\mathsf{SST}_{n}(\lambda)|}{n}$, which follows from (1), we derive that

[TABLE]

Now the assertion follows from Equation $\eqref{Eq: schur and char}$. ∎

Remark 3.4*.*

Theorem 3.3 does not hold necessarily true without the condition $\gcd(n,|\lambda|)=1$. To see this, consider the case where $n=5$, $\lambda=(2,1^{3})$ and $\Lambda:=\mathrm{wt}(\lambda)=\varpi_{1}+\varpi_{4}$. Then $\gcd(n,|\lambda|)=5\neq 1$. Since the crystal $B(\Lambda)$ is the crystal of the adjoint representation of $\mathfrak{sl}_{5}$, we have

[TABLE]

It follows from the identity c(\epsilon_{i})=\left\{\begin{array}[]{ll}\epsilon_{i+1}&\text{ if }i\neq 5,\\ \epsilon_{1}&\text{ if }i=5,\end{array}\right. that every orbit is free or consists of a singleton. One can easily see that the number of free orbits equals $4$ and the number of fixed points equals $4$. By a direct computation, we have

[TABLE]

This says that the triple $(B(\Lambda),\mathsf{C},q^{-\kappa(\lambda)}s_{\lambda}(1,q,q^{2},q^{3},q^{4}))$ does not exhibit the cyclic sieving phenomenon.

However, setting $\sigma(\lambda):=\sum_{i\in I}\frac{i(i-1)}{2}\lambda_{i}$, we can observe that

[TABLE]

Hence, quite interestingly, the triple $(B(\Lambda),\mathsf{C},q^{-\sigma(\lambda)}s_{\lambda}(1,q,q^{3},q^{6},q^{10}))$ exhibits the cyclic sieving phenomenon.

4. Commuting action with $\mathsf{c}$

Recall that $\mathsf{pr}$ is the promotion on $\mathsf{SST}_{n}(\lambda)$. For $T\in\mathsf{SST}_{n}(\lambda)$, let

[TABLE]

Proposition 4.1**.**

Let $\lambda$ be a Young diagram with $\ell(\lambda)<n$. Suppose that $\gcd(n,|\lambda|)=1$. Then we have

1. (1)

for any $T\in\mathsf{SST}_{n}(\lambda)$, $|\mathcal{O}_{\mathsf{pr}}(T)|$ is divisible by $n$, and 2. (2)

the order of $\mathsf{pr}$ on $\mathsf{SST}_{n}(\lambda)$ is divisible by $n$.

Proof.

(1) Let $T\in\mathsf{SST}_{n}(\lambda)$ and set

[TABLE]

Since $\mathrm{wt}(\mathsf{c}(T))=\mathrm{wt}(\mathsf{pr}(T))=c(\mathrm{wt}(T))$ and $n$ is the order of $c$, by Lemma 3.2, we see that

[TABLE]

Since $|\mathcal{O}_{\mathsf{pr}}(T)|\in\mathcal{T}$ by definition, we have the assertion.

(2) It follows from (1) directly. ∎

Lemma 4.2**.**

Let $\lambda$ be a Young diagram with $\ell(\lambda)<n$. Suppose that $\lambda$ is of hook shape or two-column shape. Then $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}$.

Proof.

To begin with, let us fix necessary notations for the proof.

For $k\in\mathbb{Z}_{>0}$ and $l\in\mathbb{Z}_{\geq 0}$, let ${k}^{l}:=(\underbrace{k,\ldots,k}_{l})$. For $\mathbf{i}=(i_{1},\ldots,i_{l})\in\mathbb{Z}^{l}$, let $\mathbf{i}^{+t}:=(i_{1}+t,\ldots,i_{l}+t)$, and we simply draw

$\mathbf{i}$

(resp. 

$\mathbf{i}$

) for the one-row (resp. one-column) tableau with entries $(i_{1},\ldots,i_{l})$.

For $T\in\mathsf{SST}_{n}(\lambda)$, we write $k\in T$ if $k$ appears in $T$ as an entry. For $1\leq k\leq n$, we set $T_{\leq k}$ to be the tableau obtained from $T$ by removing all boxes with entries in $\{k+1,\ldots,n\}$. We also define $T_{<k}$, $T_{\geq k}$ and $T_{>k}$ in a similar manner.

(Hook shape case)

We assume that $\lambda$ is of hook shape, and choose any $T\in\mathsf{SST}_{n}(\lambda)$. We denote by $c_{1}(T)$ (resp. $r_{1}(T)$) the first column (resp. the first row) of $T$. It is obvious that $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}(T)=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}(T)$ when $\mathsf{sh}(T_{\leq n-2})=\emptyset$. Thus we assume that $\mathsf{sh}(T_{\leq n-2})\neq\emptyset$. Let

[TABLE]

(Case 1) Suppose that $n-1,n\notin c_{1}(T)$. Then we can write $T$ and $\mathsf{s}_{n-1}(T)$ as follows:

[TABLE]

By a direct computation, we can see that

[TABLE]

which verifies the assertion since $\mathsf{s}_{1}$ exchanges the number of $1$ and $2$.

(Case 2) Suppose that $n-1\in c_{1}(T)$, but $n\notin c_{1}(T)$. We first consider the case where $y=0$. Then $T$ and $\mathsf{s}_{1}(T)$ can be written as follows:

[TABLE]

Thus we have

[TABLE]

which justifies the assertion as before.

In case where of $y\neq 0$, we can see that

[TABLE]

and thus

[TABLE]

as required.

(Case 3) Suppose that $n-1\notin c_{1}(T)$, but $n\in c_{1}(T)$. Then $T$ is given as follows:

[TABLE]

If $x=0$, then we have

[TABLE]

as required.

If $x\neq 0$, then we obtain

[TABLE]

and thus

[TABLE]

as required.

(Case 4) Suppose that $n-1,n\in c_{1}(T)$. Then $T$ and $\mathsf{s}_{n-1}(T)$ can be written as follows:

[TABLE]

A direct computation yields that

[TABLE]

as required.

(Two-columns shape case)

We assume that $\lambda$ is of two-column shape and $T\in\mathsf{SST}_{n}(\lambda)$. Let

[TABLE]

If $p=q$, then there is nothing to prove since $\mathsf{s}_{n-1}(T)=T$ and $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}(T)=\mathsf{pr}^{2}(T)$. From now on, suppose that $p\neq q$. Then we have the following cases:

[TABLE]

(Case 1) Suppose that $(p,q)=(2,0)$ or $(0,2)$. Then $T$ can be written as follows:

[TABLE]

where $a=n-1$ or $n$. Applying $\mathsf{pr}^{2}$ to $T$, we have

[TABLE]

where $b=1$ or $2$, respectively. This shows that $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}(T)=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}(T)$.

(Case 2) Suppose that $(p,q)=(0,1)$ or $(1,0)$. We first consider the case where $(p,q)=(0,1)$. Then we can write $T$ as follows:

[TABLE]

In either case, the equality $T_{<n-1}=(\mathsf{s}_{n-1}(T))_{<{n-1}}$ holds. Thus, it is easy to see that

[TABLE]

which implies that $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}(T)=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}(T)$.

The remaining case where $(p,q)=(1,0)$ can be proved in the same manner.

(Case 3) Suppose that $(p,q)=(2,1)$ or $(1,2)$. We first consider the case where $(p,q)=(2,1)$. Then $T$ can be written as follows:

[TABLE]

Then we have

[TABLE]

respectively. By the same argument as in (Case 1) and (Case 2), we have

[TABLE]

Thus, we have that $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}(T)=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}(T)$.

The remaining case where $(p,q)=(1,2)$ can be proved in the same manner. ∎

Remark 4.3*.*

It should be remarked that the identity $\mathsf{s}_{1}\cdot\mathsf{pr}^{2}=\mathsf{pr}^{2}\cdot\mathsf{s}_{n-1}$ is not true in general. Let us consider the case where $n=4$, $\lambda=(3,2,1)$ and

[TABLE]

Then

[TABLE]

Remark 4.4*.*

It should also be remarked that $\tilde{f}_{1}\cdot\mathsf{pr}^{2}\neq\mathsf{pr}^{2}\cdot\tilde{f}_{n-1}$ even in the case of a hook shape or a two-column shape (see [1, Proof of Proposition 3.2]). For example, we consider the case where $n=3$ and T=\tiny\begin{tabular}[]{|c|c|}\hline\cr 1&2\\ \hline\cr 2\\ \cline{1-1}\cr\end{tabular}\ . Then it is easy to see that

[TABLE]

Lemma 4.5**.**

Let $\lambda$ be a Young diagram with $\ell(\lambda)<n$. Suppose that $\lambda$ is of hook shape or two-column shape. Then we have

[TABLE]

In particular, we have $\mathsf{c}\cdot\mathsf{pr}^{n}=\mathsf{pr}^{n}\cdot\mathsf{c}.$

Proof.

It was shown in [1, Proposition 3.2] that $\tilde{f}_{i+1}\cdot\mathsf{pr}=\mathsf{pr}\cdot\tilde{f}_{i}$ and $\tilde{e}_{i+1}\cdot\mathsf{pr}=\mathsf{pr}\cdot\tilde{e}_{i}$ for $i=1,\ldots,n-2$. For $T\in\mathsf{SST}_{n}(\lambda)$, we have

[TABLE]

Then, it follows from the definition $\eqref{Eq: def of si}$ that

[TABLE]

By Lemma 4.2, we have

[TABLE]

which completes the proof. ∎

In the following, we assume that

[TABLE]

Let $\mathsf{P}$ be the cyclic group generated by $\mathsf{pr}^{n}$ acting on $\mathsf{SST}_{n}(\lambda)$. Then the product group $\mathsf{C}\times\mathsf{P}$ acts on $\mathsf{SST}_{n}(\lambda)$. For $T\in\mathsf{SST}_{n}(\lambda)$, we set

[TABLE]

For an $n$-tuple $\alpha\in\mathbb{Z}_{\geq 0}^{n}$, let $\mathsf{SST}_{n}(\lambda,\alpha):=\{T\in\mathsf{SST}_{n}(\lambda)\mid\mathrm{cont}(T)=\alpha\}$. We denote by $\mathrm{cont}(\lambda)$ the set of all contents of $T$ where $T$ varies over $\mathsf{SST}_{n}(\lambda)$, and by $\mathrm{cont}^{+}(\lambda)$ the set of all $\alpha=(a_{1},\ldots,a_{n})\in\mathrm{cont}(\lambda)$ such that $a_{1}\geq a_{2}\geq\cdots\geq a_{n}$. Notice that $\mathsf{SST}_{n}(\lambda,\alpha)$ is invariant under $\mathsf{pr}^{n}$ for any $\alpha\in\mathrm{cont}(\lambda)$. For clarity, denote by $\mathsf{pr}^{n}|_{\alpha}$ the restriction of $\mathsf{pr}^{n}$ to $\mathsf{SST}_{n}(\lambda,\alpha)$.

Theorem 4.6**.**

Suppose that $\eqref{Eq: assumption}$ holds. Then the following holds.

1. (1)

For $T\in\mathsf{SST}_{n}(\lambda)$, $|\mathcal{O}_{\mathsf{pr}}(T)|=|\mathcal{O}_{\mathsf{C}\times\mathsf{P}}(T)|$. 2. (2)

For an $n$-tuple $\alpha\in\mathbb{Z}_{\geq 0}^{n}$ with $\mathsf{SST}_{n}(\lambda,\alpha)\neq\emptyset$, let $\mathfrak{o}_{\lambda}(\alpha)$ be the order of $\mathsf{pr}^{n}|_{\alpha}$. Then the order of $\mathsf{pr}$ on $\mathsf{SST}_{n}(\lambda)$ equals $n\cdot\mathrm{lcm}\{\mathfrak{o}_{\lambda}(\alpha)\mid\alpha\in\mathrm{cont}^{+}(\lambda)\}$, where $\mathrm{lcm}\{k_{1},\ldots,k_{t}\}$ denotes the least common multiple of $k_{1},\ldots,k_{t}$.

Proof.

(1) Since $|\mathcal{O}_{\mathsf{C}}(T)|=n$ and the order of $\mathsf{pr}^{n}$ is given by $\displaystyle\frac{|\mathcal{O}_{\mathsf{pr}}(T)|}{n}$ by Theorem 3.3 together with Proposition 4.1, we deduce that

[TABLE]

(2) By the assumption $\eqref{Eq: assumption}$, we have that $\mathfrak{o}_{\lambda}(\alpha)=\mathfrak{o}_{\lambda}(\mathsf{s}_{i}\cdot\alpha)$ for $i\in I$. Thus, by (1), we have that

[TABLE]

∎

Example 4.7**.**

We consider the case where $n=6$ and $\lambda=(2,2,2,1)$. Then $\eqref{Eq: assumption}$ holds by Lemma 4.5 and $\mathrm{cont}^{+}(\lambda)=\{\alpha_{1}:=(2,2,2,1,0,0),\ \alpha_{2}:=(2,2,1,1,1,0),\alpha_{3}:=(2,1,1,1,1,1)\}$. As $|\mathsf{SST}_{6}(\lambda,\alpha_{1})|=1$, it follows that $\mathfrak{o}_{\lambda}(\alpha_{1})=1$. In the case of $\mathsf{SST}_{6}(\lambda,\alpha_{2})$, we have

[TABLE]

which tells us that the order $\mathfrak{o}_{\lambda}(\alpha_{2})=2$. Finally, we can see that $\mathsf{SST}_{6}(\lambda,\alpha_{3})$ is decomposed into the following two orbits:

[TABLE]

Thus $\mathfrak{o}_{\lambda}(\alpha_{3})=6$, and by Theorem 4.6, the order of $\mathsf{pr}$ is given by $6\cdot\mathrm{lcm}\{1,2,6\}=36$.

We now focus on the hook shape $\lambda=(N-m,1^{m})$. In this case, a closed formula for the order of $\mathsf{pr}$ was given in [2].

Theorem 4.8** ([2, Theorem 3.9]).**

For a hook shape $\lambda=(N-m,1^{m})$, the order of $\mathsf{pr}$ on $\mathsf{SST}_{n}(\lambda)$ is given as follows:

[TABLE]

Suppose that $\gcd(n,N)=1$. Let $\alpha=(a_{1},\ldots,a_{n})\in\mathbb{Z}_{\geq 0}^{n}$ and let $\mathsf{m}(\alpha)$ denote the number of nonzero entries in $\alpha$. It was proved in [2] that the order of $\mathsf{pr}^{n}|_{\alpha}$ is given as

[TABLE]

and the triple $(\mathsf{SST}_{n}(\lambda,\alpha),\mathsf{pr}^{n}|_{\alpha},X(q))$ exhibits the cyclic sieving phenomenon, where X(q)=\left[\begin{array}[]{c}\mathsf{m}(\alpha)-1\\ m\end{array}\right]_{q} is the $q$-binomial coefficient. For $\lambda,\mu\vdash N$, let $m_{\lambda}(x_{1},x_{2},\ldots,x_{n})$ be the monomial symmetric polynomial assocoated to $\lambda$ and let $K_{\lambda,\mu}(q)$ be the Kostka-Foulkes polynomial associated with $\lambda$ and $\mu$ (see [12] for the definitions). The following lemma is needed for the bicyclic sieving phenomenon on $\mathsf{SST}_{n}(\lambda)$, which can be proved straightforwardly.

Lemma 4.9**.**

Let $\varphi:\widetilde{C}\rightarrow C$ be a surjective homomorphism between finite cyclic groups. Suppose that the triple $(X,C,f(q))$ exhibits the cyclic sieving phenomenon. We set $d:=|\widetilde{C}|/|C|$. Then the triple $(X,\widetilde{C},f(q^{d}))$ also exhibits the cyclic sieving phenomenon via the homomorphism $\varphi$.

We now have the following bicyclic sieving phenomenon.

Theorem 4.10**.**

Let $\lambda=(N-m,1^{m})$ with $\gcd(n,N)=1$, and let $\mathsf{d}$ and $\mathsf{d}_{\alpha}$ be the orders of $\mathsf{pr}^{n}$ and $\mathsf{pr}^{n}|_{\alpha}$ respectively. We set

[TABLE]

where $A_{\mu}=\frac{\mathsf{d}}{\mathsf{d}_{\mu}}\left(-\kappa(\mu)+m\cdot\mu_{1}^{\prime}-\frac{m(m+1)}{2}\right)$ and $\kappa(\mu)$ is defined in $\eqref{Eq: schur and char}$. Then the triple $(\mathsf{SST}_{n}(\lambda),\mathsf{C}\times\mathsf{P},S_{\lambda}(q,t))$ exhibits the bicyclic sieving phenomenon.

Proof.

Let $X$ be a finite set on which a finite group $G$ acts. For $g\in G$, let $X^{g}:=\{x\in X\mid x=g\cdot x\}$ and let $o(g)$ be the order of $g$. Note that the symmetric group $\mathfrak{S}_{n}$ acts on $\mathbb{Z}_{\geq 0}^{n}$ by place permutation, i.e., $s_{i}\cdot(a_{1},\ldots,a_{n})=(a_{s_{i}(1)},\ldots,a_{s_{i}(n)})$ for $i=1,\ldots,n-1$.

Let $c:=s_{1}\cdots s_{n-1}\in\mathfrak{S}_{n}$ and choose any $\mu=(\mu_{1},\ldots,\mu_{n})\vdash N$ with $\mathsf{SST}_{n}(\lambda,\mu)\neq 0$. Let $W(\mu):=\{w\cdot\mu\mid w\in\mathfrak{S}_{n}\}$ and set $l:=\mathsf{m}(\mu)$. Note that $\mathsf{m}(\mu)=\mu_{1}^{\prime}$. It follows from Lemma 3.2 and Theorem 3.3 that

[TABLE]

For $w=s_{i_{1}}\cdots s_{i_{l}}\in\mathfrak{S}_{n}$, we set $\mathsf{w}:=\mathsf{s}_{i_{1}}\cdots\mathsf{s}_{i_{l}}$ and define

[TABLE]

and

[TABLE]

Note that $\mathsf{m}(\mu)=\mathsf{m}(w\cdot\mu)$, $\mathsf{w}\cdot\mathsf{SST}_{n}(\lambda,\mu)=\mathsf{SST}_{n}(\lambda,w\cdot\mu)$ and $|W(\mu)|=|\mathcal{O}_{\mathfrak{S}_{n}}(T)|$ for $T\in\mathsf{SST}_{n}(\lambda,\mu)$. Then the group $\mathsf{C}\times\mathsf{P}$ clearly acts on $S(\mu)$. It follows from $\eqref{Eq: distinct}$ together with Lemma 4.5 that

[TABLE]

But, as every $\mathsf{C}$-orbit of $\mathsf{SST}(\lambda)$ is free by Theorem 3.3, we can deduce that

[TABLE]

Let

[TABLE]

Note that the triple $(\mathsf{SST}_{n}(\lambda,\mu),\mathsf{pr}^{n}|_{\mu},X_{\mu}(t))$ exhibits the cyclic sieving phenomenon by [2, Theorem 4.3]. By the same argument as in the proof of Theorem 3.3, for any $T\in\mathsf{SST}_{n}(\lambda,\mu)$, one can show that $(\mathcal{O}_{\mathfrak{S}_{n}}(T),\mathsf{C},Y_{\mu}(q))$ also exhibits the cyclic sieving phenomenon. For $k\in\mathbb{Z}_{\geq 0}$, let $\omega_{k}$ be a primitive $k$th root of unity. By the definition, we have

[TABLE]

Here, the second equality for $Y_{\mu}$ follows from the fact that every $\mathsf{C}$-orbit is free. Thus, by putting Lemma 4.5, $\eqref{Eq: S}$ and $\eqref{Eq: X Y}$ together, we can derive that

[TABLE]

which tells us that the triple $(S(\mu),\mathsf{C}\times\mathsf{P},X_{\mu}(t)\cdot Y_{\mu}(q))$ exhibits the bicyclic sieving phenomenon. Since $\mathsf{SST}_{n}(\lambda)=\bigcup_{\mu\vdash N}S(\mu)$, we conclude that $(\mathsf{SST}_{n}(\lambda),\mathsf{C}\times\mathsf{P},\sum_{\mu\vdash N}X_{\mu}(t^{{\mathsf{d}}/{\mathsf{d}_{\mu}}})\cdot Y_{\mu}(q))$ exhibits the bicyclic sieving phenomenon by Lemma 4.9. Now the assertion follows from the equality $X_{\mu}(t)=t^{-\kappa(\mu)+m\cdot\mu_{1}^{\prime}-\frac{m(m+1)}{2}}K_{\lambda,\mu}(t)$ ([11, Example 4.2] or [10, Lemma 7.12]). ∎