Crystals, semistandard tableaux and cyclic sieving phenomenon

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.


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 ∈ Z >0 , let ω d be a dth primitive root of the unity. We say that (X, C, f (q)) exhibits the cyclic sieving phenomenon if, for all c ∈ C, we have 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: 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 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].
In the present paper, we investigate the cyclic sieving phenomenon on SST n (λ) with a cyclic action arising from its crystal structure (see Section 1 for crystals). For this purpose, we first notice that pr = σ 1 σ 2 · · · σ n−1 , where σ i is the ith BenderKnuth involution acting on SST n (λ). In general, σ i 's do not satisfy braid relations. We then note that SST n (λ) has a U q (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 c := s 1 s 2 · · · s n−1 on SST n (λ), where s i are simple reflections in the Weyl group. The operator c shares several similarities with pr, for instance, it is easy to check that wt(c(T )) = wt(pr(T )) = s 1 s 2 · · · s n−1 (wt(T )). One of the most favorable features of c, compared with pr, might be that its order is given by n for arbitrary shape λ, whereas the order of pr is very difficult to compute.
In the viewpoint of crystal theory, by using the operator c instead of pr, we observe a new cyclic sieving phenomenon on SST n (λ) beyond rectangular shape. More precisely, we prove that if λ is a Young diagram with ℓ(λ) < n and gcd(n, |λ|) = 1, then the triple SST n (λ), C, q −κ(λ) s λ (1, q, . . . , q n−1 ) exhibits the cyclic sieving phenomenon, where C is the cyclic group generated by c (see Theorem 3.3). There are several examples for which our cyclic sieving phenomenon hold without the condition gcd(n, |λ|) = 1, and Remark 3.4 shows an example for another cyclic sieving phenomenon with a specialization of s λ other than the principal specialization. It would be an interesting problem to give a characterization of Young diagrams λ such that (SST n (λ), c, f (q)) exhibits a cyclic sieving phenomenon, where f (q) is a suitable specialization of s λ (multiplied by a q-power). We also remark that the cyclic sieving phenomenon on the set of isolated vertices of a tensor product B ⊗m of a crystal B with a different cyclic operator was studied in [20].
We here deal with the case where λ is of hook shape or two-column shape. In these special cases, we show that pr n commutes with s i 's, thus pr n commutes with c. We then show that the order of pr n on SST n (λ) equals lcm{o λ (α) | α ∈ cont + (λ)}, where o λ (α) denotes the order of pr n | α and lcm{k 1 , k 2 , . . . , k t } the least common multiple of k 1 , k 2 , . . . , k t . We next consider the bicyclic sieving phenomenon on SST n (λ) in case where λ is of hook shape with (n, |λ|) = 1 (see [16,Section 9] for the definition). Let λ = (N − m, 1 m ) with gcd(n, N ) = 1, and consider the polynomial given in Theorem 4.10. Here m µ (x 1 , x 2 , . . . , x n ) is the monomial symmetric polynomial assocoated to µ, and K λ,µ (t) is the Kostka-Foulkes polynomial associated with λ and µ. Note that the evaluation S λ (q, t) at t = 1 is equal to q −κ(λ) s λ (1, q, . . . , q n−1 ). We show that the triple (SST n (λ), C×P, S λ (q, t)) exhibits the bicyclic sieving phenomenon, where P is the cyclic group generated by 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 c on SST n (λ) and prove the triple SST n (λ), C, q −κ(λ) s λ (1, q, . . . , q n−1 ) exhibits the cyclic sieving phenomenon. In Section 4, we investigate a connection between c and pr and show the bicyclic sieving phenomenon given by c and pr n for hook shape.

Crystals
Let I be a finite index set. A square matrix A = (a ij ) i,j∈I is called a generalized Cartan matrix if it satisfies (i) a ii = 2 for i ∈ I and a ij ∈ Z ≤0 for i = j, (ii) a ij = 0 if and only if a ji = 0, (iii) there exists a diagonal matrix D = diag(d i | i ∈ I) such that DA is symmetric. A Cartan datum (A, P, Π, P ∨ , Π ∨ ) consists of (1) a generalized Cartan matrix A, (2) a free abelian group P, called the weight lattice, (3) Π = {α i | i ∈ I} ⊂ P, called the set of simple roots, (4) P ∨ = Hom Z (P, Z), called the coweight lattice, Π is linearly independent over Q, (3) for each i ∈ I, there exists ̟ i ∈ P, called the fundamental weight, such that h j , ̟ i = δ j,i for all j ∈ I.
We set Q := i∈I Zα i , called the root lattice, and Q + := i∈I Z ≥0 α i . We fix a nondegenerate symmetric bilinear form (· , ·) on h * := Q ⊗ Z P satisfying Let us denote by P + := {λ ∈ P | h i , λ ≥ 0 for all i ∈ I} the set of dominant integral weights, and define ht(β) := i∈I k i for β = i∈I k i α i ∈ Q + . Let W be the Weyl group associated with A, which is generated by Let U q (g) be the quantum group associated with the Cartan datum (A, P, P ∨ Π, Π ∨ ), which is generated by f i , e i (i ∈ I) and q h (h ∈ 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.
For a dominant integral weight Λ ∈ P + , we denote by B(Λ) the crystal of the irreducible highest weight U q (g)-module V q (Λ) with highest weight Λ. For i ∈ I, we define the bijection s i on B(Λ) by Then the Weyl group W acts on the crystal B(Λ) in which the simple reflection s i acts via s i for i ∈ I (see [3, Chapter 2.5] for details). Note that where |B(Λ) ξ | is the number of elements of B(Λ) ξ , and e ξ are formal basis elements of the group algebra Q[P] with the multiplication given by e ξ e ξ ′ = e ξ+ξ ′ . The q-dimension of B(Λ) is given by where ev : Q → Z is the map defined as follows: We now assume that I = {1, 2, . . . , r} and the Cartan matrix A is of finite type. Note that the crystal B(Λ) is a finite set. We define the bijection c on B(Λ) as follows: Since s i 's act on B(Λ) as simple reflections of the Weyl group W, c can be viewed as a Coxeter element of W. Let C := c be the cyclic subgroup of W generated by c, and h the Coxeter number of W.
Lemma 1.2. The cyclic group C has order h and acts on the crystal B(Λ).

Semistandard tableaux
For a partition λ = (λ 1 ≥ λ 2 ≥ . . . ≥ λ l > 0), the length ℓ(λ) of λ is defined to be the number of positive parts of λ and the size |λ| of λ the sum of all parts, that is, ℓ(λ) = l and |λ| = Σλ i . Throughout this paper, we will confuse λ with its Young diagram drawn in English convention, more precisely, an array of boxes in which the ith row has λ i boxes from top to bottom. The conjugate λ ′ of λ denotes the Young diagram obtained from λ by flipping the diagonal.
A semistandard tableau T of shape λ with entries bounded by n is a filling of boxes of λ with entries in {1, 2, . . . , n} such that (1) the entries in each row are weakly increasing from left to right, and (2) the entries in each column are strictly increasing from top to bottom.
Let sh(T ) denote the shape of a semistandard tableau T and SST n (λ) the set of all semistandard tableaux of shape λ with entries bounded by n. We say that b = (p, q) ∈ T if b is a box of T at the pth row and the qth column, and denote by T (b) the entry of the box b. For example, the following is a semistandard tableau of shape λ = (8, 5, 2) with entries bounded by 5: For T ∈ SST n (λ), the content cont(T ) of T is defined to be the n-tuple (c 1 , . . . , c n ), where c k is the number of occurrences of k in T . Setting x T := x c 1 1 · · · x cn n , we define the Schur polynomial Next, we describe the promotion operator pr on SST n (λ). Let T ∈ SST n (λ). If T does not contain entries equal to n, then 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 pr(T ).
From now on, we assume that the Cartan matrix A is of type A n−1 , i.e., U q (g) = U q (sl n ), with I = {1, 2, . . . , n − 1}. For k = 1, . . . , n, we set ǫ k := (0, . . . , 1, . . . , 0) ∈ Q n to be the unit vector with the 1 in the kth position. For i ∈ I, we set Then we identify the weight lattice P with the n − 1-dimensional subspace of Q n orthogonal to the vector ǫ 1 + · · · + ǫ n . Note that the bilinear form (· , ·) corresponds to the usual inner product and s i (ǫ j ) = ǫ s i (j) for i ∈ I, where the subscript s i denotes the simple transposition It is well-known that SST n (λ) admits a U q (sl n )-crystal structure and SST n (λ) ≃ B(wt(λ)) as a U q (sl n )-crystal. We refer the reader to [3, Chapter 3] and [6, Chapter 7] for details. Note that wt(T ) = c 1 ǫ 1 + · · · + c n ǫ n for T ∈ SST n (λ), where cont(T ) = (c 1 , . . . , c n ). We remark that the principal specialization of s λ (x 1 , . . . , x n ) is equal to the q-dimension of B(wt(λ)) up to a power of q, more precisely, Since SST n (λ) is a U q (sl n )-crystal, the operator c defined as in (1.4) acts on SST n (λ). The lemma below follows from Lemma 1.2 immediately.
Lemma 2.2. The cyclic group C has order n and acts on the U q (sl n )-crystal SST n (λ).

Cyclic sieving phenomenon
As before, assume that the Cartan matrix A is of type A n−1 . Let c := s 1 s 2 · · · s n−1 ∈ S n . Note that S n acts on the weight lattice P. In addition, from the definition of pr and c it follows that wt(c(T )) = wt(pr(T )) = c(wt(T )) for T ∈ SST n (λ). (1) For β ∈ Q, we have ev(c(β)) ≡ ev(β) (mod n).
Proof. (1) As ev is linear, it suffices to consider the case where β = α i for i ∈ I. By a direct computation, we can derive that This tells us that ev(c(α i )) ≡ ev(α i ) (mod n).
For positive integers a, b ∈ Z >0 , we denote by gcd(a, b) the greatest common divisor of a and b. A subset {a 1 , a 2 , . . . , a n } ⊂ Z is called a complete residue system modulo n if it has no two elements that are congruent modulo n.

Then we have
(1) for any T ∈ SST n (λ), |O pr (T )| is divisible by n, and (2) the order of pr on SST n (λ) is divisible by n.
Since wt(c(T )) = wt(pr(T )) = c(wt(T )) and n is the order of c, by Lemma 3.2, we see that Since |O pr (T )| ∈ T by definition, we have the assertion.
Lemma 4.2. Let λ be a Young diagram with ℓ(λ) < n. Suppose that λ is of hook shape or two-column shape. Then s 1 · pr 2 = pr 2 · s n−1 .
Proof. To begin with, let us fix necessary notations for the proof. For k ∈ Z >0 and l ∈ Z ≥0 , let k l := (k, . . . , k l ). For i = (i 1 , . . . , i l ) ∈ Z l , let i +t := (i 1 + t, . . . , i l + t), and we simply draw i (resp. i ) for the one-row (resp. onecolumn) tableau with entries (i 1 , . . . , i l ). For T ∈ SST n (λ), we write k ∈ T if k appears in T as an entry. For 1 ≤ k ≤ n, we set T ≤k to be the tableau obtained from T by removing all boxes with entries in {k + 1, . . . , n}. We also define T <k , T ≥k and T >k in a similar manner.

(Hook shape case)
We assume that λ is of hook shape, and choose any T ∈ SST n (λ). We denote by c 1 (T ) (resp. r 1 (T )) the first column (resp. the first row) of T . It is obvious that s 1 · pr 2 (T ) = pr 2 · s n−1 (T ) when sh(T ≤n−2 ) = ∅. Thus we assume that sh(T ≤n−2 ) = ∅. Let x := the number of occurrences of n − 1 in r 1 (T ), y := the number of occurrences of n in r 1 (T ).
(Case 1) Suppose that n − 1, n / ∈ c 1 (T ). Then we can write T and s n−1 (T ) as follows: By a direct computation, we can see that , which verifies the assertion since s 1 exchanges the number of 1 and 2.
(Case 2) Suppose that n − 1 ∈ c 1 (T ), but n / ∈ c 1 (T ). We first consider the case where y = 0. Then T and s 1 (T ) can be written as follows: Thus we have , which justifies the assertion as before. In case where of y = 0, we can see that and thus as required.
(Case 4) Suppose that n − 1, n ∈ c 1 (T ). Then T and s n−1 (T ) can be written as follows: A direct computation yields that , as required.
(Two-columns shape case) We assume that λ is of two-column shape and T ∈ SST n (λ). Let p := the number of occurrences of n − 1 in T , q := the number of occurrences of n in T .
Then pr 2 · s 3 (T ) = . Then it is easy to see that Lemma 4.5. Let λ be a Young diagram with ℓ(λ) < n. Suppose that λ is of hook shape or two-column shape. Then we have s i · pr n = pr n · s i for i ∈ I.
In particular, we have c · pr n = pr n · c.
In the following, we assume that (4.1) gcd(n, |λ|) = 1 and s i · pr n = pr n · s i for i ∈ I.
Let P be the cyclic group generated by pr n acting on SST n (λ). Then the product group C × P acts on SST n (λ). For T ∈ SST n (λ), we set For an n-tuple α ∈ Z n ≥0 , let SST n (λ, α) := {T ∈ SST n (λ) | cont(T ) = α}. We denote by cont(λ) the set of all contents of T where T varies over SST n (λ), and by cont + (λ) the set of all α = (a 1 , . . . , a n ) ∈ cont(λ) such that a 1 ≥ a 2 ≥ · · · ≥ a n . Notice that SST n (λ, α) is invariant under pr n for any α ∈ cont(λ). For clarity, denote by pr n | α the restriction of pr n to SST n (λ, α).
We now focus on the hook shape λ = (N − m, 1 m ). In this case, a closed formula for the order of pr was given in [2]. Suppose that gcd(n, N ) = 1. Let α = (a 1 , . . . , a n ) ∈ Z n ≥0 and let m(α) denote the number of nonzero entries in α. It was proved in [2] that the order of pr n | α is given as and the triple (SST n (λ, α), pr n | α , X(q)) exhibits the cyclic sieving phenomenon, where X(q) = m(α) − 1 m q is the q-binomial coefficient. For λ, µ ⊢ N , let m λ (x 1 , x 2 , . . . , x n ) be the monomial symmetric polynomial assocoated to λ and let K λ,µ (q) be the Kostka-Foulkes polynomial associated with λ and µ (see [12] for the definitions). The following lemma is needed for the bicyclic sieving phenomenon on SST n (λ), which can be proved straightforwardly.
Lemma 4.9. Let ϕ : C → 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 := | C|/|C|. Then the triple (X, C, f (q d )) also exhibits the cyclic sieving phenomenon via the homomorphism ϕ.
Proof. Let X be a finite set on which a finite group G acts. For g ∈ G, let X g := {x ∈ X | x = g · x} and let o(g) be the order of g. Note that the symmetric group S n acts on Z n ≥0 by place permutation, i.e., s i · (a 1 , . . . , a n ) = (a s i (1) , . . . , a s i (n) ) for i = 1, . . . , n − 1.