Crystals and Schur $P$-positive expansions

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.


Introduction
Let P + be the set of strict partitions and let P λ be the Schur P -function corresponding to λ ∈ P + [12]. The set of Schur P -functions is an important class of symmetric functions, which is closely related with representation theory and algebraic geometry (see [10] and references therein). For example, the Schur P -polynomial P λ (x 1 , . . . , x n ) in n variables is the character of a finite-dimensional irreducible representation V n (λ) of the queer Lie superalgebra q(n) with highest weight λ up to a power of 2 when the length ℓ(λ) of λ is no more than n [13].
The set of Schur P -functions forms a basis of a subring of the ring of symmetric functions, and the structure constants with respect to this basis are non-negative integers, that is, given λ, µ, ν ∈ P + , for some non-negative integers f λ µν . The first and the most well-known result on a combinatorial description of f λ µν was obtained by Stembridge [16] using shifted Young tableaux, which is a combinatorial model for Schur P -or Q-functions [11,17]. It is shown that f λ µν is equal to the number of semistandard tableaux with entries in a Z 2graded set N = { 1 ′ < 1 < 2 ′ < 2 < · · · } of shifted skew shape λ/µ and weight ν such that (i) for each integer k ≥ 1 the southwesternmost entry with value k is unprimed or of even degree and (ii) the reading words satisfy the lattice property. Here we say that the value |x| is k when x is either k or k ′ in a tableau. Let us call these tableaux the Littlewood-Richardson-Stembridge (LRS) tableaux (Definitions 3.8 and 3.9).
Recently, two more descriptions of f λ µν were obtained in terms of semistandard decomposition tableaux, which is another combinatorial model for Schur P -functions introduced by Serrano [14]. It is shown by Cho that f λ µν is given by the number of semistandard decomposition tableaux of shifted shape µ and weight w 0 (λ − ν) whose reading words satisfy the λ-good property (see [3,Corollary 5.14]). Here we assume that ℓ(λ), ℓ(µ), ℓ(ν) ≤ n, and w 0 denotes the longest element in the symmetric group S n . Another description is given by Grantcharov, Jung, Kang, Kashiwara, and Kim [6] based on their crystal base theory for the quantized enveloping algebra of q(n) [7]. They realize the crystal B n (λ) associated to V n (λ) as the set of semistandard decomposition tableaux of shape λ with entries in { 1 < 2 < · · · < n }, and describe f λ µν by characterizing the lowest weight vectors of weight w 0 λ in the tensor product B n (µ) ⊗ B n (ν). We also remark that bijections between the above mentioned combinatorial models for f λ µν are studied in [4] using insertion schemes for semistandard decomposition tableaux.
The main result in this paper is to give another description of f λ µν using the theory of q(n)-crystals, and show that it is indeed equivalent to that of Stembridge. More precisely, we show that f λ µν is equal to the number of semistandard tableaux with entries in N of shifted skew shape λ/µ and weight ν such that (i) for each integer k ≥ 1 the southwesternmost entry with value k is unprimed or of even degree and (ii) the reading words satisfy the "lattice property" (see Definitions 3.3 and 3.4 and Theorem 3.5). It is obtained by semistandardizing the standard tableaux which parametrize the lowest weight vectors counting f λ µν in [6], where the "lattice property" naturally arises from the configuration of entries in semistandard decomposition tableaux. We show that these tableaux for f λ µν are equal to LRS tableaux (Theorem 3.11), and hence obtain a new characterization of LRS tableaux.
We study other Schur P -or Q-positive expansions and their combinatorial descriptions from a viewpoint of crystals. First we consider the Schur P -positive expansion of a skew Schur function s λ/δr = ν∈P + a λ/δr ν P ν for a skew diagram λ/δ r contained in a rectangle ((r + 1) r+1 ), where δ r = (r, r − 1, . . . , 1) [1]. We give a combinatorial description of a λ/δr ν (Theorem 4.4) by considering a q(n)-crystal structure on the set of usual semistandard tableaux of shape λ/δ r and characterizing the lowest weight vectors corresponding to each ν ∈ P + .
As a byproduct we also give a simple alternate proof of Ardila-Serrano's description of a λ/δr ν [1] (Theorem 4.7), which can be viewed as a standardization of our description.
We next consider the Schur expansion of a Schur P -function is a super Schur function in variables x and y. We give a simple and alternate proof of Stembridge's description of g λµ [16] (Theorem 5.1) by characterizing the type A lowest weight vectors of weight w 0 µ in the q(n)-crystal B n (λ) when ℓ(λ), ℓ(µ) ≤ n. Finally, we introduce the notion of semistandard decomposition tableaux of shifted skew shape. We consider a q(n)-crystal structure on the set of such tableaux, and describe its decomposition into B n (λ)'s, which implies that the corresponding character has a Schur P -positive expansion though it is not equal to a skew Schur P -function in general.
The paper is organized as follows. In Section 2, we review the notion of q(n)crystals and related results. In Section 3, we describe a combinatorial description of f λ µν and show that it is equivalent to that of Stembridge. In Sections 4 and 5, we discuss the Schur P -positive expansion of a skew Schur function and the Schur expansion of a Schur P -function, respectively. In Section 6, we discuss semistandard decomposition tableaux of shifted skew shape, and the Schur P -positive expansions of their characters.
2. Crystals for queer Lie superalgebras 2.1. Notation and terminology. In this subsection, we introduce necessary notations and terminologies. Let Z + be the set of non-negative integers. We fix a positive integer n ≥ 2 throughout this paper. Let be the set of strict partitions. For λ ∈ P, let ℓ(λ) denote the length of λ, and |λ| = i≥1 λ i . Let P n = { λ | ℓ(λ) ≤ n } ⊆ P and P + n = P + ∩ P n . The (unshifted) diagram of λ ∈ P is defined to be the set and the shifted diagram of λ ∈ P + is defined to be the set We identify D λ and D + λ with diagrams where a box is placed at the i-th row from the top and the j-th column from the left for each (i, j) ∈ D λ and D + λ , respectively. For instance, if λ = (6, 4, 2, 1), then Let A be a linearly ordered set. We denote by W A the set of words of finite length with letters in A. For w ∈ W A and a ∈ A, let c a (w) be the number of occurrences of a in w.
For λ, µ ∈ P with D µ ⊆ D λ , a tableau of shape λ/µ means a filling on the skew diagram D λ \D µ with entries in A. For λ, µ ∈ P + with D + µ ⊆ D + λ , a tableau of shifted shape λ/µ is defined in a similar way. For a tableau T of (shifted) shape λ/µ, let w(T ) be the word given by reading the entries of T row by row from top to bottom, and from right to left in each row. We denote by T i,j the j-th entry (from the left) of the i-th row of T from the top. For 1 ≤ i ≤ ℓ(λ), let T (i) = T i,λ i · · · T i,1 be the subword of w(T ) corresponding to the i-th row of T . Then we have w(T ) = T (1) · · · T (ℓ(λ)) . We denote by w rev (T ) the reverse word of w(T ). Note that T i,j is not the entry of T at the (i, j)-position of the (shifted) skew diagram of λ/µ, that is, For a ∈ A, let c a (T ) = c a (w(T )) be the number of occurrences of a in T .
Suppose that A is a linearly ordered set with a Z 2 -grading A = A 0 ⊔ A 1 . For λ, µ ∈ P with D µ ⊆ D λ , let SST A (λ/µ) be the set of tableaux of shape λ/µ with entries in A which is semistandard, that is, (i) the entries in each row (resp. column) are weakly increasing from left to right (resp. from top to bottom), (ii) the entries in A 0 (resp. A 1 ) are strictly increasing in each column (resp. row). Similarly, for λ, µ ∈ P + with D + µ ⊆ D + λ , we define SST + A (λ/µ) to be the set of semistandard tableaux of shifted shape λ/µ with entries in A.
where the Z 2 -grading and linear ordering are induced from N. For a ∈ N, we write |a| = k when a is either k or k ′ .

2.2.
Semistandard decomposition tableaux and Schur P -functions. Let us recall the notion of semistandard decomposition tableaux [6,14], which is our main combinatorial object. (1) A word u = u 1 · · · u s in W N is called a hook word if it satisfies u 1 ≥ u 2 ≥ · · · ≥ u k < u k+1 < · · · < u s for some 1 ≤ k ≤ s. In this case, let u↓= u 1 · · · u k be the weakly decreasing subword of maximal length and u↑= u k+1 · · · u s the remaining strictly increasing subword in u.
(2) For λ ∈ P + , let T be a tableau of shifted shape λ with entries in N. Then T is called a semistandard decomposition tableau of shape λ if For any hook word u, the decreasing part u↓ is always nonempty by definition. For λ ∈ P + , let SSDT (λ) be the set of semistandard decomposition tableaux of shape λ. Let x = {x 1 , x 2 , . . .} be a set of formal commuting variables, and let P λ = P λ (x) be the Schur P -function in x corresponding to λ ∈ P + (see [10]). It is shown in [14] that P λ is given by the weight generating function of SSDT (λ):

Remark 2.2.
Recall that the Schur P -function P λ can be realized as the character of tableaux T ∈ SST + N (λ) with no primed entry or entry of odd degree on the main diagonal (cf. [10,11,17]). The notion of semistandard decomposition tableaux was introduced in [14] to give a plactic monoid model for Schur P -functions. In this paper, we follow its modified version (Definition 2.1) introduced in [6], by which it is more easier to describe q(n)-crystals [6, Remark 2.6]. We also refer the reader to [4] for more details on relation between the combinatorics of these two models.
The following is a useful criterion for a tableau to be a semistandard decomposition one, which plays an important role in this paper. For λ ∈ P + , let T be a tableau of shifted shape λ with entries in N. Then T ∈ SSDT (λ) if and only if T (k) is a hook word for 1 ≤ k ≤ ℓ(λ), and none of the following conditions holds for each 1 ≤ k < ℓ(λ): Equivalently, T ∈ SSDT (λ) if and only if T (k) is a hook word for 1 ≤ k ≤ ℓ(λ), and the following conditions hold for 1 ≤ k < ℓ(λ): For λ ∈ P + , let SSDT n (λ) be the set of tableaux T ∈ SSDT (λ) with entries in [n]. By Proposition 2.3(1), we see that SSDT n (λ) = ∅ if and only if λ ∈ P + n . We denote by P λ (x 1 , . . . , x n ) the Schur P -polynomial in x 1 , . . . , x n given by specializing P λ at x n+1 = x n+2 = · · · = 0. Then we have P λ (x 1 , . . . , x n ) = T ∈SSDTn(λ) x T .
For λ ∈ P + n , let H λ n be the element in SSDT n (λ) where the subtableau with entry ℓ(λ) − i + 1 is a connected border strip of size λ ℓ(λ)−i+1 starting at (i, i) ∈ D + λ for each i = 1, . . . , ℓ(λ), and let L λ n be the one where the subtableau with entry n − i + 1 is a connected horizontal strip of size λ i starting at (i, i) ∈ D + λ for each i = 1, . . . , ℓ(λ). For example, when n = 4 and λ = (4, 3, 1), we have Indeed, H λ n and L λ n are the unique tableaux in SSDT n (λ) such that Here we assume that P + n ⊂ Z n + and the symmetric group S n acts on Z n + by permutation, where w 0 is the longest element in S n .

2.3.
Crystals. Let us first review the crystals for the general linear Lie algebra gl(n) in [8,9].
Let P ∨ = n i=1 Ze i be the dual weight lattice and P = Hom A gl(n)-crystal is a set B together with the maps wt : . . , n − 1 satisfying the following conditions: for b ∈ B and i = 1, . . . , n − 1, Here 0 is a formal symbol and −∞ is the smallest element in is finite for all µ, we define the character of B by chB = µ∈P |B µ |e µ , where e µ is a basis element of the group algebra Q[P ]. Let For λ ∈ P n , let B n (λ) be the crystal associated to an irreducible gl(n)-module with highest weight λ, where we regard λ as n i=1 λ i ǫ i ∈ P + . We may regard [n] as B n (ǫ 1 ), where wt(k) = ǫ k for k ∈ [n], and hence W [n] as a gl(n)-crystal where we identify w = w 1 . . . w r with w 1 ⊗ · · · ⊗ w r ∈ B n (ǫ 1 ) ⊗r . The crystal structure on W [n] is easily described by so-called the signature rule (cf. [9, Section 2.1]). For λ ∈ P n , the set SST [n] (λ) becomes a gl(n)-crystal under the identification of T with w(T ) ∈ W [n] , and it is isomorphic to B n (λ) [9]. In general, one can define a gl(n)crystal structure on SST [n] (λ/µ) for a skew diagram λ/µ. By abuse of notation, we set B n (λ/µ) := SST [n] (λ/µ).
Next, let us review the notion of crystals associated to polynomial representations of the queer Lie superalgebra q(n) developed in [6,7].
Let B n be a q(n)-crystal which is the gl(n)-crystal B n (ǫ 1 ) together with f 1 1 = 2 (in dashed arrow): For q(n)-crystals B 1 and B 2 , the tensor product B 1 ⊗ B 2 is the gl(n)-crystal B 1 ⊗ B 2 where the actions of e 1 and f 1 are given by Then it is easy to see that Suppose that B is a regular gl(n)-crystal, that is, each connected component in B is isomorphic to B n (λ) for some λ ∈ P n . Let W = S n be the Weyl group of gl(n) which is generated by the simple reflection r i corresponding to α i for i = 1, . . . , n − 1. We have a group action of W on B denoted by S such that For λ ∈ P + , let B n (λ) = SSDT n (λ), and consider an injective map Then we have the following.
n is a unique q(n)-highest weight vector and L λ n is a unique q(n)-lowest weight vector.
Remark 2.6. In [7], a semisimple tensor category over the quantum superalgebra U q (q(n)) is introduced, and it is shown that each irreducible highest weight module V n (λ) in this category, parametrized by λ ∈ P + n , has a crystal base. Furthermore, it is shown in [6, Theorem 2.5(c)] that the crystal of V n (λ) is isomorphic to B n (λ).
Let B 1 and B 2 be q(n)-crystals. For b 1 ∈ B 1 and b 2 ∈ B 2 , let us say that b 1 and b 2 are equivalent and write The following lemma plays a crucial role in characterization of q(n)-lowest weight vectors in B ⊗N n and hence describing the decompositions of B ⊗N n and B n (µ)⊗B n (ν) (µ, ν ∈ P + n ) into connected components in [6].
n , the following are equivalent: Hence, we have the following immediately by Lemma 2.7.
Remark 2.9. Let m ≥ n be a positive integer, and put t = m − n. For N ≥ 1, let

Littlewood-Richardson rule for Schur P -functions
For λ, µ, ν ∈ P + , the shifted Littlewood-Richardson coefficients f λ µν are the coefficients given by In this section we give a new combinatorial description of f λ µν using the theory of q(n)-crystals. We also show that our description of f λ µν is equivalent to the Stembridge's description [16].

Shifted Littlewood-Richardson rule.
Definition 3.1. Let w = w 1 · · · w N be a word in W N . Let m k = c k (w) + c k ′ (w) for k ≥ 1. We define w * = w * 1 · · · w * N to be the word obtained from w after applying the following steps for each k ≥ 1: (1) Consider the letters w i 's with |w i | = k. Label them with 1, 2, . . . , m k (as subscripts), first enumerating the w p 's with w p = k from left to right, and then w q 's with w q = k ′ from right to left. (2) After the step (1), remove all ′ in each labeled letter k ′ j , that is, replace Definition 3.3. Let w = w 1 · · · w N ∈ W N be given. We say that w satisfies the "lattice property" if the word w * = w * 1 · · · w * N associated to w given in Definition 3.1 satisfies the following for k ≥ 1: for some s < t and i ≥ 1, then no k + 1 j for i < j occurs in w * s · · · w * t , (L3) if (w * s , w * t ) = (k j+1 , k + 1 j ) for some s < t and j ≥ 1, then no k i for i ≤ j occurs in w * s · · · w * t .
Then we have the following characterization of f λ µν .
Theorem 3.5. For λ, µ, ν ∈ P + , we have that is, the shifted LR coefficient f λ µν is equal to the number of tableaux in F λ µν .
Let us prove f λ µν = |F λ µν | by constructing a bijection Let T ∈ L λ µν be given. Assume that w rev (T ) = u 1 · · · u N where N = |ν|. By Lemma 2.7, there exists µ (m) ∈ P + n for 1 ≤ m ≤ N such that Here we assume that µ (0) = µ. Recall that where T (k) = T k,1 · · · T k,λ k is a hook word for 1 ≤ k ≤ ℓ(ν). We define Q T to be a tableau of shifted shape λ/µ with entries in N, where µ (m) /µ (m−1) is filled with for some 1 ≤ k ≤ ℓ(ν). In other words, the boxes in Q T corresponding to T (k) ↑ are filled with k ′ from right to left as a vertical strip and then those corresponding to T (k) ↓ are filled with k from left to right as a horizontal strip. By construction, it is clear that Q T ∈ SST + N (λ/µ) with c k ′ (Q T ) + c k (Q T ) = ν k for 1 ≤ k ≤ ℓ(ν). Let w(Q T ) = w 1 · · · w N . Since T (k) is a hook word for each k and the rightmost letter, say u m , in T (k) ↓ is strictly smaller than the leftmost letter u m+1 in T (k) ↑, the entry k in Q T corresponding to u m is located to the southeast of all k ′ 's in Q T . So the conditions Definition 3.4(1) and (2) are satisfied.
It remains to check that w(Q T ) satisfies the "lattice property". Note that if we label k and k ′ in (3.5) as k j and k ′ j , respectively when u m = T k,j , then it coincides with the labeling on the letters in w(Q T ) given in Definition 3.1(1). Now it is not difficult to see that the conditions Proposition 2.3(1), (2), and (3) on T implies the conditions Definition 3.3 (L1), (L2), and (L3), respectively. Therefore, Q T ∈ F λ µν . Finally the correspondence T → Q T is injective and also reversible. Hence the map (3.4) is a bijection. This completes the proof.
Remark 3.6. We see from Remark 2.9 that |L λ µν | does not depend on n for all sufficiently large n. Hence (3.3) also implies the Schur P -positivity of the product P µ P ν . Remark 3.7. For T ∈ L λ µν , let Q T be the tableau of shifted shape λ/µ, which is defined in the same way as Q T in the proof of Theorem 3.5 except that we fill µ (m) /µ (m−1) with m in (3.5) for 1 ≤ m ≤ N . Then the set { Q T | T ∈ L λ µν } is equal to the one given in [6,Theorem 4.13] to describe f λ µν . For example, 3 Definition 3.8. Let w = w 1 · · · w N be a word in W N and w rev be the reverse word of w. Let w be the word obtained from w by replacing k by (k + 1) ′ and k ′ by k for each k ≥ 1. Suppose that w w rev = a 1 · · · a 2N , and let m k (i) = c k (a 1 · · · a i ) for k ≥ 1 and 0 ≤ i ≤ 2N . Then we say that w satisfies the lattice property if Here we assume that m k (0) = 0.
We call LRS λ µν the set of Littlewood-Richardson-Stembridge tableaux.
that is, the shifted LR coefficient f λ µν is equal to the number of tableaux in LRS λ µν .
Next, we claim that w satisfies (L2). Suppose that there is a triple (w * s , w * u , w * t ) = (k + 1 i , k + 1 j , k i+1 ) for some k ≥ 1, i < j, and 1 ≤ s < u < t ≤ N . We may assume that j = i + 1. Since w * s = k + 1 i is placed to the left of w * u = k + 1 i+1 , it follows from Definition 3.1 that a s = k + 1, and from Definition 3.9(2) that a u = k + 1. Since w satisfies the lattice property, there is a positive integer v < s such that w * v = k i , i.e., a v = k and m k (v) = i for some v < s. We have m k+1 (u − 1) = m k (u − 1) = i and a u = k + 1, a contradiction. So w satisfies (L2).
Finally, we claim that w satisfies (L3). Suppose for the sake of contradiction that (w * s , w * u , w * t ) = (k j+1 , k i , k + 1 j ) for some k ≥ 1, i ≤ j, and 1 ≤ s < u < t ≤ N . We may assume that i = j. Since w * s = k j+1 is placed to the left of w * u = k j , it follows that a s = k ′ . We consider four cases depending on the primedness of a u and a t as follows: Case 2. Let a u = k ′ and a t = k+1. It follows that a 2N −u+1 = k, m k (2N −u+1) = j and m k+1 (t) = j. Since t < 2N − u + 1, we have m k (t) < m k+1 (t). So there is an integer 0 ≤t < t such that m k (t) = m k+1 (t) < j and at +1 = k + 1, as desired.
Indeed, we have shown in the proof of Theorem 3.11 that Corollary 3.12. Let w ∈ W N be such that (1) (c k (Q) + c k ′ (Q)) k≥1 ∈ P + , (2) for k ≥ 1, if x is the rightmost letter in w with |x| = k, then x = k.
Then w satisfies the "lattice property" in Definition 3.3 if and only if w satisfies the lattice property in Definition 3.8.
Remark 3.13. A bijection from LRS λ µν to L λ µν is also given in [4,Theorem 4.7], which coincides with the inverse of the map T → Q T in (3.4) (see also the remarks in [4, p.82]). The proof of [4,Theorem 4.7] use insertion schemes for two versions of semistandard decomposition tableaux and another combinatorial model for f λ µν by Cho [3] as an intermediate object between LRS λ µν and L λ µν . On the other hand, we prove more directly that the map T → Q T in (3.4) is a bijection from L λ µν to LRS λ µν by using a new characterization of the lattice property in Theorem 3.11.
It is shown in [1,5] that the skew Schur function s λ/δ k has a non-negative integral expansion in terms of Schur P -functions (4.1) s λ/δr = ν∈P + a λ/δr ν P ν , together with a combinatorial description of a λ/δr ν . Moreover it is shown that these skew Schur functions are the only ones (up to rotation of shape by 180 • ), which have Schur P -positivity. In this section, we give a new simple description of a λ/δr ν using q(n)-crystals. First we consider a q(n)-crystal structure on B n (λ/δ r ), which is a slight generalization of [7, Example 2.10(d)].
Proposition 4.1. Let λ ∈ P n be such that D δr ⊆ D λ ⊆ D (r+1) r+1 . Then the gl(n)crystal B n (λ/δ r ) as a subset of W [n] together with 0 is invariant under e 1 and f 1 . Hence B n (λ/δ r ) is a q(n)-crystal.
Proof. Let N = |λ| − |δ r |. For T ∈ B n (λ/δ r ), let w(T ) = w 1 · · · w N . Recall that T is identified with w(T ) in W [n] . Here we call the box in D λ/δr containing w i the w i -box, and call the set of boxes (x, r − x + 2) ∈ D λ/δr for 1 ≤ x ≤ r + 1 the main anti-diagonal of D λ/δr . Suppose that f 1 w(T ) = 0. There exists 1 ≤ i ≤ N − 1 such that w i = 1 and w j = 1, 2 for all i < j ≤ N , and by the tensor product rule (2.3). We first observe that the entry 1 in T can be placed only on the main anti-diagonal in D λ/δr . If there is a box in D λ/δr below the w i -box, then it corresponds to w j for some j > i, and hence its entry is greater than 2. Moreover, if there is a box in D λ/δr to the right of the w i -box, then its entry is greater than 1 since it is not on the main anti-diagonal. So we conclude that there exists T ′ ∈ SST [n] (λ/δ r ) such that w(T ′ ) = f 1 w(T ).
Suppose that e 1 w(T ) = 0. There exists 1 ≤ i ≤ N − 1 such that w i = 2 and w j = 1, 2 for all i < j ≤ N , and by the tensor product rule (2.3). If the w i -box is not on the main anti-diagonal, then the w i+1 -box is placed to the left of the w i -box. Then the w i+1 -box is filled with 1 or 2, which contradicts (4.2). So the w i -box is on the main anti-diagonal, and thus f 1 w(T ) = w(T ′ ) for some T ′ ∈ B n (λ/δ r ). This completes the proof. Proof. Since B n (λ/δ r ) is a q(n)-crystal, the skew Schur polynomial s λ/δr (x 1 , . . . , x n ) is a non-negative integral linear combination of P ν (x 1 , . . . , x n ). Then we apply Remark 2.9.
Note that w rev (T ) = T (r+1) · · · T (1) , where T (l) = T l,1 · · · T l,λ l −r−1+l is a weakly increasing word corresponding to the l-th row of T for 1 ≤ l ≤ r + 1. Let Q T be a tableau of shifted shape ν with entries in N, where ν (m) /ν (m−1) is filled with r + 2 − l if u m occurs in T (l) , for some 1 ≤ l ≤ r +1. Note that the boxes in Q T corresponding to T (l) are filled with r + 2 − l as a horizontal strip. So Q T satisfies the condition Definition 4.3 (1).
For each k ≥ 1, let us enumerate the letter k's in Q T from southwest to northeast like k 1 , k 2 , . . .. Since T ∈ SST n (λ/δ r ), we see that the entry k i in Q T corresponds to T l,i for i ≥ 1, where l = r + 2− k, and moreover (k + 1) i is located in the southwest of k i+1 for i ≥ 2. This implies the condition Definition 4.3 (2), and hence Q T ∈ A λ/δr ν .
Finally, one can check that correspondence T → Q T is a bijection.
Let N = |δ r+1 | − |µ|, and let T δ r+1 /µ be the tableau obtained by filling δ r+1 /µ with 1, 2, . . . , N subsequently, starting from the bottom row to top, and from left to right in each row. For instance, For ν ∈ P + with |ν| = N , let B δ r+1 /µ ν be the set of tableaux Q such that where each entry i ∈ [N ] occurs exactly once, (2) if j is directly above i in T δ r+1 /µ , then j is placed strictly to the right of i in Q, (3) if i + 1 is placed to the right of i in T δ r+1 /µ , then i + 1 is strictly below i in Q.
We define Q ′ T to be the tableau of shifted shape ν such that ν (m) /ν (m−1) is filled with m for 1 ≤ m ≤ N . Then we have the following.
Proof. Let T ′ λ/δr be the tableau obtained by filling λ/δ r with 1, 2, . . . , N subsequently, starting from the leftmost column to rightmost, and from bottom to top in each column. For instance, when λ = (5,4,4,4,2) and r = 4, we have where each entry i ∈ [N ] occurs exactly once, (2) if j is directly above i in T ′ λ/δr , then then j is strictly below i in Q ′ T , (3) if i + 1 is placed to the right of i in T ′ λ/δr , then i + 1 is placed strictly to the right of i in Q ′ T . We see that T δ r+1 /(λ c ) ′ is obtained from T ′ λ/δr by flipping with respect to the main anti-diagonal. This implies that Q ′ Corollary 4.8. Under the above hypothesis, we have a bijection Recall that for a skew shape η/ζ, we have s η/ζ = s (η/ζ) π , where (η/ζ) π is the (skew) diagram obtained from η/ζ by rotating 180 degree (which can be seen for example by reversing the linear ordering on N in [2]). Also if s η/ζ has a Schur P -expansion, then we have s η/ζ = s η ′ /ζ ′ by applying the involution ω on the ring symmetric function sending s η to s η ′ since ω(P ν ) = P ν for ν ∈ P + (see [10, p.  Hence we have for λ ∈ P such that D δr ⊆ D λ ⊆ D ((r+1) r+1 ) . This implies that for ν ∈ P + , where a λ/δr ν are given in (4.1). Equivalently, we have for µ ∈ P with D µ ⊆ D δ r+1 . Therefore Theorem 4.6 follows from Theorem 4.4, Corollary 4.8, and (4.6) (or (4.7)).

Schur expansion of Schur P -function
For λ ∈ P + and µ ∈ P, let g λµ be the coefficient of s µ in the Schur expansion of P λ , that is, The purpose of this section is to give an alternate proof of the following combinatorial description of g λµ due to Stembridge.
where G λµ is the set of tableaux Q such that w(Q) satisfies the lattice property.
Proof. The proof is similar to that of Theorem 3.5. Choose n such that λ ∈ P + n and µ ∈ P n . Let Then we have as a gl(n)-crystal and hence g λµ = |L λµ | by linear independence of Schur polynomials. Let us define a map Since T is a gl(n)-lowest weight vector, we have by (2.2) that u N −m+1 ⊗ · · · ⊗ u N ∈ B ⊗m n is a gl(n)-lowest weight element for 1 ≤ m ≤ N . This implies that there exists µ (m) ∈ P n for 1 ≤ m ≤ N such that u N −m+1 · · · u N is equivalent as an element of gl(n)-crystal to a gl(n)-lowest weight element in B n (µ (m) ), where µ (N ) = µ and µ (m) is obtained by adding a box in the (n − u m + 1)-st row of µ (m−1) with µ (0) = ∅.
We define Q T to be a tableau of shape µ with entries in N, for some 1 ≤ k ≤ ℓ(λ). By almost the same arguments as in the proof of Theorem 3.5, we see that Q T satisfies the conditions (1) and (2) for G λµ , and w(Q T ) satisfies the "lattice property", which implies that it satisfies the lattice property by Corollary 3.12. (We leave the details to the reader.) Finally the correspondence T → Q T is a well-defined bijection.
Remark 5.3. Let λ ∈ P + be such that D + λ ⊆ D + δ r+1 for some r ≥ 0. Let λ c+ be a strict partition obtained by counting complementary boxes D + δ r+1 \ D + λ in each column from right to left. It is shown in [5] that By (4.6) or (4.7), we have g ν λ = a λ c /δr (ν c+ ) ′ . One may expect that there is a natural bijection between G ν λ and A λ c /δr (ν c+ ) ′ , but we do not know the answer yet.
Let T be a tableau of shifted skew shape λ/µ. For p, q ≥ 1, let T (p, q) denote the entry of T at the p-th row and the q-th diagonal from the main diagonal in D + λ (that is, { (i, i) | i ≥ 1} ∩ D + λ ) whenever it is defined. Note that T (p, q) is not necessarily equal to T p,q if µ is nonempty.
Proof. Choose a sufficiently large M such that all the entries in L µ M are greater than n. For a tableau T of shifted shape λ/µ with entries in [n], let T := L µ M * T be the tableau of shifted shape λ, that is, the subtableau of shape shifted µ in T is L µ M and its complement in T is T . By definition of SSDT (λ/µ) and Proposition 2.3, we have (6.2) T ∈ B n (λ/µ) if and only if T ∈ B M (λ).
Let T ∈ B n (λ/µ) and i ∈ I be given. Ifx i T = 0 (x = e, f ), then we have by (6.2) thatx i T = L µ M * T ′ for some T ′ ∈ B n (λ/µ). This implies thatx i w rev (T ) = w rev (T ′ ). Therefore, the image of B n (λ/µ) in (6.1) together with {0} is invariant under the action of e i and f i for i ∈ I.
By applying Theorem 3.5 repeatedly, we see that f λ/µ ν for ν ∈ P + n in this case is equal to the number of tableaux Q such that (1) Q ∈ SST + N (ν) with c k (Q) + c k ′ (Q) = η k for k ≥ 1, (2) for each k ≥ 1, if x is the rightmost in w(Q) with |x| = k, then x = k.