1 Introduction

The cyclic sieving phenomenon (CSP) was introduced by Reiner et al. [13]. Given a set X, a cyclic group C of order n acting on X and a polynomial f(q), the triple (X, C, f(q)) is said to exhibit the cyclic sieving phenomenon if for all \(k \in {\mathbb {Z}}\),

$$\begin{aligned} |\{ x \in X: g^k \cdot x = x \}| = f(\xi ^k) \end{aligned}$$

where g is a generator of C, and \(\xi \) is a primitive nth root of unity. To this date, there many different instances of the cyclic sieving phenomenon has been proved; see for example, the survey by Sagan [14].

Let \(\textrm{SSYT}(\lambda , n)\) denote the set of skew semi-standard Young tableaux with shape \(\lambda \) and maximal entry at most n. Note that this set is empty unless \({{\,\mathrm{\ell }\,}}(\lambda )\le n\). We also set \(n(\lambda ):= \sum _{j} (j-1) \lambda _j\). The Schur polynomial \(\textrm{s}_\lambda (\textbf{x})\) is defined as

$$\begin{aligned} \textrm{s}_\lambda (x_1,\dotsc ,x_n):= \sum _{T \in \textrm{SSYT}(\lambda ,n)} \textbf{x}^T \end{aligned}$$

where \(\textbf{x}^T\) is the monomial \(x_1^{w_1(T)} \cdots x_n^{w_n(T)}\) and \(w_i(T)\) is the number of entries in T equal to i; see [9, 16] for more background. The symmetric group \(S_n\) act on \(\textrm{SSYT}(\lambda ,n)\) via type A crystal reflection operators; see Kashiwara and Nakashima [7] and Shimozono’s [15] survey for an excellent background. The book [4] provides a more thorough introduction.

Oh and Park [11] show that the triple

$$\begin{aligned} \left( \textrm{SSYT}(\lambda ,n), \langle c_n \rangle , q^{-n(\lambda )}\textrm{s}_\lambda (1,q,q^2,\dotsc ,q^{n-1})\right) \end{aligned}$$
(1.1)

exhibits the cyclic sieving phenomenon, where \(c_n = \tilde{\textrm{s}}_1 \tilde{\textrm{s}}_2\cdots \tilde{\textrm{s}}_{n-1}\) is a product of crystal reflection operators. Note that \(c_n\) has order n. In this note, we generalize (1.1) to skew shapes, and refine the statement to smaller sets of tableaux; see Theorem 3.4 and Corollary 3.5. The CSP in (1.1) is in stark contrast with the conjectured CSP in [1], where we conjecture that there is a cyclic group C of order n, acting on \(\textrm{SSYT}(n\lambda ,m)\) and

$$\begin{aligned} \left( \textrm{SSYT}(n\lambda ,m), C, \textrm{s}_{n\lambda }(1,q,q^2,\dotsc ,q^{m-1})\right) \end{aligned}$$
(1.2)

is a CSP-triple (with no restriction on m). Here, \(n\lambda \) denotes element-wise multiplication \((n\lambda _1,\dotsc ,n\lambda _\ell )\). This conjecture was recently proved in [12, Cor. 4.4] using crystals. Notably, (1.2) does not extend to skew shapes in general, and it is an open problem to characterize for which combinations of skew shapes, n and m for which (1.2) holds. However, some notable skew cases have been resolved; see [8, Thm. 1, Thm. 2].

A SYT analog of (1.2) is proved in [3, Thm. 49], where we show that there is a group C of order n acting on the set of skew standard Young tableaux of shape \(n\lambda /n\mu \), such that

$$\begin{aligned} \left( \textrm{SYT}(n\lambda /n\mu ), C, f^{n\lambda /n\mu }(q)\right) \end{aligned}$$

is a CSP-triple. Here, \(f^{n\lambda /n\mu }(q)\) is the major-index generating polynomial for skew SYT.

2 Preliminaries

Let \(\textrm{WCOMP}(m,n)\) denote the set of weak compositions, which is the set of vectors \((\alpha _1,\dotsc ,\alpha _n)\) in \({\mathbb {N}}_{\ge 0}^n\) such that \(\alpha _1+\alpha _2+\cdots +\alpha _n = m\). Given a composition \(\alpha \), we let \(|\alpha |\) denote the sum of the entries. We use the same notation for integer partitions, and we let \(|\lambda /\mu |\) denote the difference \(|\lambda |-|\mu |\) whenever \(\lambda /\mu \) is a skew shape.

Given a weak composition \(\alpha \), let \(\textrm{SSYT}(\lambda /\mu ,\alpha )\) denote the subset of \(\textrm{SSYT}(\lambda /\mu , n)\) where each tableau has exactly \(\alpha _i\) entries equal to i. Note that this set is non-empty only if \(|\lambda /\mu |=|\alpha |\). If \(\mu = \emptyset \), we simply write \(\textrm{SSYT}(\lambda ,\alpha )\), and \(|\textrm{SSYT}(\lambda ,\alpha )|\) is the Kostka coefficient \(K_{\lambda ,\alpha }\). The skew Schur polynomial \(\textrm{s}_{\lambda /\mu }(\textbf{x})\) is defined analogously to the Schur polynomials, as a sum over skew SSYT:

$$\begin{aligned} \textrm{s}_{\lambda /\mu }(x_1,\dotsc ,x_n):= \sum _{T \in \textrm{SSYT}(\lambda /\mu ,n)} \textbf{x}^T = \sum _{\nu \vdash |\lambda /\mu | } K_{\lambda /\mu ,\nu } \textrm{m}_{\nu }(x_1,\dotsc ,x_n). \end{aligned}$$

For a weak composition \(\alpha = (\alpha _1,\dotsc ,\alpha _n)\), let \({{\,\textrm{cyc}\,}}_r(\alpha )\) denote the cyclic shift \((\alpha _{1+r},\alpha _{2+r},\dotsc ,\alpha _{n+r})\) where indices are computed modulo n. In particular, \(\alpha = {{\,\textrm{cyc}\,}}_n(\alpha )\).

Lemma 2.1

Let \(\alpha \in \textrm{WCOMP}(m,n)\) and suppose that \(\gcd (m,n)=1\). Then, all cyclic shifts \({{\,\textrm{cyc}\,}}_1(\alpha )\), \({{\,\textrm{cyc}\,}}_2(\alpha )\), \(\dotsc \), \({{\,\textrm{cyc}\,}}_n(\alpha )\) are different.

Proof

Suppose two cyclic shifts of \(\alpha \) are equal. Then,

$$\begin{aligned} \alpha = (a_1,\dotsc ,a_l,\; a_1,\dotsc ,a_l,\; \dotsc \; a_1,\dotsc ,a_l) \end{aligned}$$

where \(kl=n\) and \(k>1\). Then, \(k(a_1+\cdots +a_l)=m\) so it follows that k|m. But then \(k | \gcd (m,n)\), a contradiction. \(\square \)

Corollary 2.2

Let \(\lambda /\mu \) be a skew shape and \(\alpha \in \textrm{WCOMP}(|\lambda /\mu |,n)\) such that \(\gcd (|\lambda /\mu |,n)=1\). Then, the sets \(\{ \textrm{SSYT}(\lambda /\mu , {{\,\textrm{cyc}\,}}_r(\alpha )) \}_{r=1}^n\) are pairwise disjoint.

Lemma 2.3

Let \(\alpha \in \textrm{WCOMP}(m,n)\) such that \(\gcd (m,n)=1\). Then, \(z_1,z_2,\dotsc ,z_n\) defined via \(z_r:= \sum _{j=1}^n j \alpha _{j+r}\) for \(r=1,2,\dotsc ,n\) are all different mod n.

Proof

It is easy to see that for all \(r=1,2,\dotsc ,n-1\), we have the relation \(z_{r+1} \equiv z_r + m \mod n\). Since \(\gcd (m,n)=1\), the statement now follows from standard group theory. \(\square \)

Corollary 2.4

Let \(\lambda /\mu \) be a skew shape and \(\alpha \in \textrm{WCOMP}(|\lambda /\mu |,n)\) such that \(\gcd (|\lambda /\mu |,n)=1\). Then,

$$\begin{aligned} \sum _{r=1}^n \sum _{T \in \textrm{SSYT}(\lambda /\mu ,{{\,\textrm{cyc}\,}}_r(\alpha ))} q^{w_1(T)+w_2(T)+\cdots + w_n(T)} \end{aligned}$$
(2.1)

is a multiple of \([n]_q = 1+q+q^2+\cdots + q^{n-1}\) \(\mod q^n-1\).

Proof

By unraveling the definitions, (2.1) is equal to

$$\begin{aligned} \sum _{r=1}^n |\textrm{SSYT}(\lambda /\mu ,{{\,\textrm{cyc}\,}}_r(\alpha ))| q^{\alpha _{1+r} + 2\alpha _{2+r} + \cdots + n \alpha _{n+r}} \end{aligned}$$

where indices are taken modulo n. Since the cardinality of \(\textrm{SSYT}(\lambda /\mu ,{{\,\textrm{cyc}\,}}_r(\alpha ))\) does not depend on the ordering of the entriesFootnote 1 in \(\alpha \), the expression (2.1) is equal to

$$\begin{aligned} |\textrm{SSYT}(\lambda /\mu ,\alpha )| \sum _{r=1}^n q^{\alpha _{1+r} + 2\alpha _{2+r} + \cdots + n \alpha _{n+r}}. \end{aligned}$$

By Lemma 2.3, the sum is now equal to \([n]_q\) mod \((q^n-1)\), which proves the assertion.

\(\square \)

3 Crystals and Main Proofs

We have that \(S_n\) act on \(\textrm{SSYT}(\lambda /\mu , n)\) via type A crystal reflection operators, see definition in Shimozono’s survey [15]. We briefly sketch the definition here.

Definition 3.1

Let \(T \in \textrm{SSYT}(\lambda /\mu )\) be a skew Young tableau. The crystal reflection operator \(\tilde{\textrm{s}}_i\) acts on T as follows. First, consider only the entries i and \(i+1\) in T; these entries constitute a sub-tableau \(T'\) of T with at most two entries in each column. If a column of \(T'\) contains two entries, these must be i and \(i+1\). By reading the columns from left to right, ignoring columns with 0 or 2 entries, we obtain a word w in the letters i and \(i+1\). We now treat each entry i as a right parenthesis and each \(i+1\) as a left parenthesis. Entries which are matched under normal bracketing rules are now ignored, and we are left with a subword of w of the form \(i^a \; (i+1)^b\) with \(a,b \ge 0\). By replacing some is with \((i+1)\)s or vice versa, this subword can uniquely be turned into the subword \(i^b \; (i+1)^a\). Finally, \(\tilde{\textrm{s}}_i(T)\) is obtained from T by performing the same replacements as in the subword.

We omit the proof that the result is indeed also a semi-standard Young tableau, and that the \(\tilde{\textrm{s}}_i\) indeed define an \(S_n\) action on \(\textrm{SSYT}(\lambda /\mu , n)\); see [4] for details.

Example 3.2

For example, with \(i=2\) and T as follows:

we obtain the subword \(w=33 2 2233 2233 33\) (bold entries in T). After bracketing, we get the subword 23333, and this is turned into 22223 by \(\tilde{\textrm{s}}_2\). Hence, \(\tilde{\textrm{s}}_2(T)\) is given by

where the bold entries now indicate the difference between T and \(\tilde{\textrm{s}}_2(T)\).

The simple reflection \(\tilde{\textrm{s}}_i \in S_n\) is an involution

$$\begin{aligned} \tilde{\textrm{s}}_i : \textrm{SSYT}(\lambda /\mu ,\alpha ) \rightarrow \textrm{SSYT}(\lambda /\mu , s_i \alpha ), \end{aligned}$$
(3.1)

where \(s_i\) act on \(\alpha \) by interchanging entries at positions i and \(i+1\). Let \(c_n:= \tilde{\textrm{s}}_1 \tilde{\textrm{s}}_2\cdots \tilde{\textrm{s}}_{n-1} \in S_n\), and let \(\alpha \in \textrm{WCOMP}(|\lambda /\mu |,n)\). One can then show that \(c_n\) gives a bijection

$$\begin{aligned} c_n : \textrm{SSYT}(\lambda /\mu ,\alpha ) \rightarrow \textrm{SSYT}(\lambda /\mu ,{{\,\textrm{cyc}\,}}_1(\alpha )) \end{aligned}$$
(3.2)

and \(\langle c_n \rangle \subseteq S_n\) is a cyclic group of order n. These properties of \(c_n\) in (3.2) are the only properties of crystal operators that we shall use in this note. Note that the \(c_n\)-action is different from promotion obtained as a product of Bender–Knuth involutions, as promotion does not have order n in general.

Example 3.3

Let \(\lambda /\mu = 322/1\) and \(\alpha = (2,1,2,1)\). Then,

are two orbits under \(c_4\). Note that \(\gcd (6,4)=2\) here, so Corollary 2.2, and Lemma 2.3, Corollary 2.4 do not apply.

Theorem 3.4

Let \(\lambda /\mu \) be a skew shape with \(m = |\lambda /\mu |\) and \(\alpha \in \textrm{WCOMP}(|\lambda /\mu |,n)\) such that \(\gcd (m,n)=1\). Furthermore, let X be the (disjoint) union

$$\begin{aligned} X = \bigcup _{r=1}^n \textrm{SSYT}(\lambda /\mu ,{{\,\textrm{cyc}\,}}_r(\alpha )). \end{aligned}$$

Then, the triple

$$\begin{aligned} \left( X, \langle c_n \rangle , \sum _{T \in X} q^{w_1(T)+w_2(T)+\cdots + w_n(T)}\right) \end{aligned}$$

exhibits the cyclic sieving phenomenon.

Proof

By Corollary 2.2, it follows that every \(c_n \)-orbit in X has size n. By Corollary 2.4, we have that \(f(q):= \sum _{T \in X} q^{w_1(T)+w_2(T)+\cdots + w_n(T)}\) is a multiple of \([n]_q\). It follows that for all \(k \in {\mathbb {Z}}\),

$$\begin{aligned} |\{ T \in X: (c_n)^k \cdot T = T \}| = f(\xi ^k) \end{aligned}$$

where \(\xi \) is a \(n\textrm{th}\) root of unity so the triple is indeed a CSP-triple. \(\square \)

Corollary 3.5

Let \(\gcd (|\lambda /\mu |,n)=1\). Then, the triple

$$\begin{aligned} \left( \textrm{SSYT}(\lambda /\mu ,n), \langle c_n \rangle , \textrm{s}_{\lambda /\mu }(1,q,\dotsc ,q^{n-1}) \right) \end{aligned}$$

exhibits the cyclic sieving phenomenon. Moreover, if \(\mu =\emptyset \),

$$\begin{aligned} \left( \textrm{SSYT}(\lambda ,n), \langle c_n \rangle , q^{-n(\lambda )} \textrm{s}_{\lambda }(1,q,\dotsc ,q^{n-1}) \right) \end{aligned}$$
(3.3)

is also a CSP-triple.

Proof

By summing over all possible compositions \(\alpha \) in \(\textrm{WCOMP}(|\lambda /\mu |,n)\) in Theorem 3.4, we immediately have that

$$\begin{aligned} \left( \textrm{SSYT}(\lambda /\mu ,n), \langle c_n \rangle , \textrm{s}_{\lambda /\mu }(q,q^2,\dotsc ,q^n) \right) \end{aligned}$$

is a CSP-triple and \(\textrm{s}_{\lambda /\mu }(q,q^2,\dotsc ,q^n)\) evaluates to 0 for all \(q=\xi ,\xi ^2,\dotsc ,\xi ^{n-1}\) whenever \(\xi \) is a primitive \(n^\textrm{th}\) root of unity. It is also straightforward to see that \(\textrm{s}_{\lambda /\mu }(q,q^2,\dotsc ,q^n) = q^{|\lambda /\mu |} \textrm{s}_{\lambda /\mu }(1,q,\dotsc ,q^{n-1}), \) so \(\textrm{s}_{\lambda /\mu }(1,q,\dotsc ,q^{n-1})\) must evaluate to 0 for all \(q=\xi ,\xi ^2,\dotsc ,\xi ^{n-1}\) as well. Since entries in row j in a non-skew semi-standard Young tableau have value at least j, \(q^{-n(\lambda )}\textrm{s}_{\lambda }(1,q,\dotsc ,q^{n-1})\) is a polynomial in q. These observations prove the two statements. \(\square \)

We can actually say something much more general, where a cyclic sieving phenomenon can be constructed for any homogeneous, Schur-positive symmetric function.

Corollary 3.6

Let \(f(\textbf{x}) = \sum _{T \in X} \textbf{x}^T\) be a homogeneous symmetric function of degree m, such that the Schur expansion \( f(\textbf{x}) = \sum _{\lambda \vdash m} d_{\lambda } \textrm{s}_\lambda (\textbf{x}) \) is non-negative with \(d_{\lambda } \in {\mathbb {N}}\). Furthermore, suppose \(\gcd (m,n)=1\) and that there is a bijection \(\psi \):

$$\begin{aligned} \psi : X \rightarrow \bigcup _\lambda [d_\lambda ]\times \textrm{SSYT}(\lambda ,n). \end{aligned}$$

Moreover, suppose that there is a cyclic group action \(\langle c_n \rangle \) of order n act on X in such a way that it acts (via \(\psi \)) as the product of crystal reflections \(c_n\) on \(\textrm{SSYT}(\lambda ,n)\). Then,

$$\begin{aligned} \left( X, \langle c_n \rangle , f(1,q,q^2,\dotsc ,q^{n-1}) \right) \end{aligned}$$

is a CSP-triple, and every orbit under \(c_n\) has size n.

Note that such a bijection \(\psi \) and action \(c_n\) can in principle be found for any Schur-positive symmetric function. In particular, we can use the Littlewood–Richardson rule on \(\textrm{s}_{\lambda /\mu }(x_1,\dotsc ,x_n)\), to express it as a non-negative sum of non-skew Schur polynomials and then apply previous corollary.

The previous corollary can then be applied to the following families of symmetric functions, whose Schur expansion can be proved by an explicit bijection \(\psi \), and there is a type A crystal structure on the underlying combinatorial objects: Modified Hall–Littlewood symmetric functions, [6], type A and Stanley symmetric functions, [10], type C Stanley symmetric functions, [5], specialized non-symmetric Macdonald polynomials; see [2], and (some) dual k-Schur functions, [10].