Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretical proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.


Introduction
Various types of tableaux are studied in combinatorics and representation theory. Representation theory provides powerful tools to prove theorems concerning tableaux and helpful clues to generalize them. The aim of this article is to give a representation-theoretical proof of generalizations and variants of refined Burrill conjecture for oscillating tableaux. These generalizations and variants are equinumerous results between generalized oscillating tableaux and semistandard tableaux.
(ii) The diagram of λ (i) is obtained from that of λ (i−1) by adding or removing one cell for each i.
Here l(λ) is the length of a partition λ, and c(λ) is the number of columns of odd length in the diagram of λ.
In this paper, we present generalizations and variants of Theorem 1.1 from the view point of representation theory of classical groups. Our main results in the symplectic group case can be stated as follows. (See Theorem 5.3 for similar results in the orthogonal group case.) We remark that Krattenthaler [7,Theorem 4] gave a bijective proof to part (1) of the following theorem.
See Sections 2 and 5 for undefined terminologies.
Here we give an outline of our proof of Theorem 1.2. Let V = C 2n be the defining representation of the general linear group GL 2n = GL 2n (C), which is also the defining representation of the symplectic group Sp 2n = Sp 2n (C). We consider the tensor products S α (V ) = S α 1 (V ) ⊗ · · · ⊗ S α k (V ) and α (V ) = α 1 (V ) ⊗ · · · ⊗ α k (V ) of the symmetric and exterior powers of V . We compute in two ways the multiplicities of the irreducible Sp 2n -module V (m) corresponding to the one-row partition (m) in S α (V ) and α (V ).
One of the key ingredients of the proof is the Pieri rules for classical groups, which are another contribution of this paper. The Pieri rule (resp. dual Pieri rule) for Sp 2n describes the irreducible decomposition of the tensor product of an irreducible representation with S r (V ) (resp. r (V )). By iteratively applying the Pieri rule given in Theorem 3.2 (1) (resp. the dual Pieri rule in Theorem 4.1 (1)), we see that the multiplicity of V (m) in S α (V ) (resp. α (V )) is equal to the number of the sequences of partitions described in (a) of Theorem 1.2 (1) (resp. (2)).
On the other hand, by using the classical Pieri rule for GL 2n , we obtain the GL 2nmodule decompositions where λ runs over all partitions of length ≤ 2n, V λ is the irreducible representation of GL 2n corresponding to λ, and CSTab(λ, α) (resp. RSTab(λ, α)) denotes the set of column-strict (resp. row-strict) tableaux of shape λ and weight α. Another key ingredient of the proof is the following restriction multiplicity formula (see Theorem 5.4): It follows from (1.1) and (1.2) that the multiplicity of V (m) in S α (V ) (resp. α (V )) is equal to the number of tableaux described in (b) of Theorem 1.2 (1) (resp. (2)). In this way, we prove Theorem 1.2. The remaining of this paper is organized as follows. In Section 2 we review the representation theory of classical groups. In Sections 3 and 4, we prove the Pieri rules and the dual Pieri rules for the symplectic and orthogonal groups. In Section 5, we present variants/generalizations of Theorem 1.1 and give a representation-theoretical proof to them.

Preliminaries
In this section, first we recall some definitions on partitions. Then we review the representation theory of classical groups and collect several facts which will be used in the remaining of the paper. See [3], [4] and [9] for the representation theory of classical groups.

Combinatorics of partitions
A partition is a weakly decreasing sequence λ = (λ 1 , λ 2 , . . . ) of nonnegative integers such that i≥1 λ i is finite. The length of λ, denoted by l(λ), is the number of nonzero entries of λ, and the size of λ, denoted by |λ|, is the sum of entries of λ. A partition λ is often identified with its diagram, which is a left-justified array of |λ| cells with λ i cells in the ith row. The conjugate partition λ ′ of a partition λ is the partition whose diagram is obtained by reflecting the diagram of λ along the main diagonal. We denote by P the set of all partitions.
For two partitions λ and µ, we write µ ⊂ λ if µ i ≤ λ i for all i. Then the skew diagram λ/µ is defined to be the set-theoretical difference of the diagrams of λ and µ. The size of the skew diagram is defined by |λ/µ| = |λ| − |µ|. We say that the skew diagram λ/µ is a horizontal r-strip if it contains at most one cell in each column and |λ/µ| = r. Dually, we say that the skew diagram λ/µ is a vertical r-strip if it contains at most one cell in each row and |λ/µ| = r. Note that λ/µ is a horizontal strip if and only if λ 1 ≥ µ 1 ≥ λ 2 ≥ µ 2 ≥ . . . .

Representation theory of GL N
It is well-known that the irreducible polynomial representations of the general linear group GL N = GL N (C) are parametrized by partitions of length ≤ N . We denote by V λ and S λ the irreducible representation and its character corresponding to a partition λ respectively. If λ is a one-row partition (r) (resp. a one-column partition (1 r )), then we write H r = S (r) (resp. E r = S (1 r ) ). Note that H r (resp. E r ) is the characters of the symmetric power S r (V ) (resp. the exterior power r (V )) of the defining representation V = C N of GL N . We also use the notations V λ,GL N , S λ,GL N , H r,GL N and E r,GL N to avoid confusions. Let R(GL N ) be the representation ring of GL N . Then R(GL N ) is a free Z-module with basis Let Λ be the ring of symmetric functions. Let s λ ∈ Λ be the Schur function associated with a partition λ and h r = s (r) ∈ Λ the complete symmetric function of degree r. If π = π GL N : Λ → R(GL N ) is the ring homomorphism defined by π(h r ) = H r for r ≥ 1, then we have For partitions λ, µ and ν, we denote by LR λ µ,ν the Littlewood-Richardson coefficient, which is defined by the relation It is known that LR λ µ,ν = 0 unless |λ| = |µ| + |ν|, λ ⊃ µ and λ ⊃ ν. The following is the classical Pieri rule for GL N . Proposition 2.1. Let V be the defining representation of GL N . For a partition µ of length ≤ N and a nonnegative integer r, we have where λ (resp. ρ) runs over all partitions of length ≤ N such that λ/µ is a horizontal r-strip (resp. ρ/µ is a vertical r-strip). In other words, for partitions λ and µ and a nonnegative integer r, we have

Representation theory of Sp 2n
Next we consider the symplectic group Sp 2n = Sp 2n (C). The finite-dimensional irreducible representations of Sp 2n are indexed by partitions of length ≤ n. Let P(Sp 2n ) be the set of all partitions of length ≤ n. We denote by V λ = V λ ,Sp 2n and S λ = S λ ,Sp 2n the irreducible representation and its character of Sp 2n corresponding to a partition λ with l(λ) ≤ n. Let V = C 2n be the defining representation of Sp 2n , and denote by H r (resp. E r ) the character of Sp 2n on S r (V ) (resp. r (V )). Then S r (V ) is the irreducible representation corresponding to the one-row partition (r), while r (V ) is not irreducible if r ≥ 2 and the quotient r (V )/ r−2 (V ) is the irreducible representation corresponding to the one-column partition (1 r ). Let R(Sp 2n ) be the representation ring of Sp 2n For an arbitrary partition λ, we define the corresponding symplectic Schur function s λ ∈ Λ by putting Let π Sp 2n : Λ → R(Sp 2n ) be the ring homomorphism defined by π Sp 2n (h r ) = H r for r ≥ 1. The image of a symplectic Schur function under π Sp 2n can be expressed as a linear combination of irreducible characters (in fact, it is 0 or an irreducible character up to sign), by using the following algorithm (see [5] and [6]).
Proposition 2.2. Let λ be a partition.
We have the following relations in the ring Λ of symmetric functions, from which we can derive identities involving irreducible characters of Sp 2n by applying π Sp 2n .
(2) (Littlewood) For any partition λ, we have where E ′ is the set of all partitions whose column lengths are all even.
Let {f C r (x)} be the sequence of Laurent polynomials defined by For a sequence α = (α 1 , . . . , α n ) of integers and a sequence x = (x 1 , . . . , x n ) of indeterminates, we put , is a Laurent polynomial in x 1 , . . . , x n . Then the Weyl character formula is rephrased as follows: . (2.7)

Representation theory of O N
The finite-dimensional irreducible representations of the orthogonal group O N = O N (C) are parametrized by partitions such that the sum of the lengths of the first two columns is at most N . Let P(O N ) be the set of partitions λ satisfying λ ′ ,O N the irreducible representation and its character of O N corresponding to a partition λ ∈ P(O N ). Let V = C N be the defining representation of O N , and denote by H r (resp. E r ) the character of O N on S r (V ) (resp. r (V )).
Then r (V ) is the irreducible representation corresponding to the one-column partition (1 r ), while S r (V ) is not irreducible if r ≥ 2 and the quotient S r (V )/S r−2 (V ) is the irreducible representation corresponding to the one-row partition (r). Let R(O N ) be the representation ring of O N For an arbitrary partition λ, we define the corresponding orthogonal Schur function The image of an orthogonal Schur function under π O N can be expressed as a linear combination of irreducible characters (in fact, it is 0 or an irreducible character up to sign), by using the following algorithm (see [5] and [6]).
Proposition 2.5. Let λ be a partition.
(1) If λ ∈ P(O N ), then we have (2) In general, the image π O N s [λ] can be computed as follows. We put r = λ 1 and (a) If α has an entry larger than or equal to N + r, (c) Otherwise, suppose that α 1 > · · · > α p > N/2 ≥ α p+1 and define a sequence β by putting Let γ be the rearrangement of β in decreasing order and σ ∈ S r a permutation satisfying σ(β) = γ. If µ is the partition given by γ We have the following relations in the ring of symmetric functions, from which we can derive identities involving irreducible characters of O N by applying π O N . Proposition 2.6. (1) (Newell, Littlewood) For any partitions µ and ν, we have (2) (Littlewood) For any partition λ, we have where E is the set of all partitions whose row lengths are all even.
Finally we review the representation theory of the special orthogonal group SO N . We associate to a partition λ ∈ P(O N ) another partition λ ♯ ∈ P(O N ) obtained by replacing the first column (of length λ ′ 1 ) by the column of length N − λ ′ 1 . Then we have ,SO 2n+1 : λ ∈ P, l(λ) ≤ n} forms a complete set of representatives of isomorphism classes of irreducible representations of SO 2n+1 . The even orthogonal group case is more subtle. It is known that the irreducible representations of SO 2n are parametrized by sequences ω = (ω 1 , . . . , ω n−1 , ω n ) of integers satisfying ω 1 ≥ · · · ≥ ω n−1 ≥ |ω n |. We denote the corresponding irreducible representation by L [ω],SO 2n . If λ is a partition of length < n, then the restriction ,SO 2n is not irreducible and decomposes into the direct sum of two distinct irreducible representations L [λ + ],SO 2n and For a sequence α = (α 1 , . . . , α n ) of integers and a sequence x = (x 1 , . . . , x n ) of indeterminates, we put Then the Weyl character formula is rephrased as follows: (1) if X ∈ SO 2n+1 has the eigenvalues x 1 , . . . , x n , x −1 1 , . . . , x −1 n and 1, then we have . (2.11) (2) If X ∈ SO 2n has the eigenvalues x 1 , . . . , x n , x −1 1 , . . . , x −1 n , then we have . (2.12)

Pieri rules for the classical groups
In this section, we give the Pieri rules for Sp 2n and O N , which describe the irreducible decomposition of the tensor product of an irreducible representation with the symmetric power of the defining representation. The proof uses the symplectic and orthogonal Schur functions and their specialization algorithms At the level of symplectic and orthogonal Schur functions, we have the following "universal" Pieri rule.
By applying the homomorphisms π Sp 2n to (3.1) and π O N to (3.2), and then by using the algorithms given in Propositions 2.2 and 2.5, we can prove the following "actual" Pieri rules. Part (1) of the following theorem was obtained by Sundaram [10,Theorem 4.1], where she used the Berele insertion algorithm to give a combinatorial proof.
(1) Let λ, µ ∈ P(Sp 2n ) and r a nonnegative integer. Then the mul- where V is the defining representation of Sp 2n , is equal to the number of partitions ξ satisfying the following two conditions: (i) µ/ξ and λ/ξ are both horizontal strips.
(2) Let λ, µ ∈ P(O N ) and r a nonnegative integer. Then the multiplicity of the irre- is equal to the number of partitions ξ satisfying the following three conditions: (i) µ/ξ and λ/ξ are both horizontal strips.
In the orthogonal group case, the symmetric power S r (V ) of the defining representation V of O N is decomposed as follows: where ⌊r/2⌋ is the largest integer not exceeding r/2. Hence we have (ii) |µ/ξ| + |λ/ξ| = r − 2s for some integer 0 ≤ s ≤ r/2.
Also we have the following Pieri rules for the special orthogonal groups. Part (1) of the following corollary was given in [11,Theorem 5.3].
,SO 2n+1 is equal to the number of partitions ξ satisfying the following three conditions: (i) µ/ξ and λ/ξ are both horizontal strips.
In Theorem 3.2 and Corollary 3.4, we specialize r = 1 to obtain the following decomposition of the tensor product with the defining representation. Corollary 3.6. (1) If λ ∈ P(Sp 2n ) and V is the defining representation of Sp 2n , then we have where the direct sum is taken over all λ ∈ P(Sp 2n ) such that the diagram of λ is obtained from that of µ by adding or removing one cell.
(2) If λ ∈ P(O N ) and V is the defining representation of O N , then we have where the direct sum is taken over all λ ∈ P(O N ) such that the diagram of λ is obtained from that of µ by adding or removing one cell.
(3) If λ is a partition of length ≤ n and V is the defining representation of SO 2n+1 , then we have where λ runs over all partitions of length ≤ n satisfying one of the following three conditions: (i) λ ⊃ µ and |λ| = |µ| + 1.
(4) If λ is a partition of length ≤ n and V is the defining representation of SO 2n , then we have where λ runs over all partitions of length ≤ n and m(λ, µ, n) is given by (3.4).

Dual Pieri rules for classical groups
In this section, we give the dual Pieri rules for the classical groups, which describe the irreducible decomposition of the tensor product of an irreducible representation with the exterior power of the defining representation. We use the Weyl character formulas to obtain the following dual Pieri rules. (1) Let µ, λ be partitions of length ≤ n and r an integer with 0 ≤ r ≤ 2n. The multiplicity of the irreducible Sp 2n -module V λ in the tensor product V µ ,Sp 2n ⊗ r (V ), where V is the defining representation of Sp 2n , is equal to the number of partitions ξ satisfying the following three conditions: (i) l(ξ) ≤ n.
(2) Let µ, λ be partitions of length ≤ n and r an integer with 0 ≤ r ≤ 2n. The multiplicity of the irreducible SO 2n+1 module V [λ],SO 2n+1 in the tensor product where V is the defining representation of SO 2n+1 , is equal to the number of partitions ξ satisfying the following four conditions: (ii) ξ/µ and ξ/λ are both vertical strips.
(3) For two partitions µ and λ of length ≤ n and an integer r with 0 ≤ r ≤ 2n, let K λ µ,r (n) be the number of partitions ξ satisfying the following four conditions: (i) l(ξ) ≤ n.
Then we have where E r,SO 2n is the character of the exterior power r (V ) of the defining representation V of SO 2n , λ runs over all partitions of length ≤ n and m(λ, µ, n) is given by (3.4).
Remark 4.2. Part (1) of Theorem 4.1 was given in [10,Theorem 4.4]. For the special orthogonal groups, Sundaram [11,Theorem 5.4] and Weyman [12, Theorems B n and D n ] gave similar dual Pieri formulas. It is also possible to apply the generalized Littlewood-Richardson rule [8] to obtain dual Pieri rules, but the resulting formulas look more complicated than the formulas presented in Theorem 4.1.
(2) Let X ∈ SO 2n+1 have the eigenvalues x 1 , . . . , x n , x −1 1 , . . . , x −1 n , 1. By using the Weyl character formula (2.11) and we have We note that First we consider the case where l(µ) = n. In this case, by the same argument as in the proof of (1), we have where λ runs over all partitions of length ≤ n. Hence we have Next we consider the case where l = l(µ) < n. Let v r and w r be the column vectors given by Then we have det w µ 1 +n−1 · · · w µ l +n−l w n−l−1 · · · w 1 w 0 .
If l(λ) < l, then tha maps are injective and If l(ρ) = l and 0 ≤ s ≤ n − l, then the maps µ,r+n−l+s (n) are injective and Combining these observations with (4.1) completes the proof of (2).
(3) Let X ∈ SO 2n have the eigenvalues x 1 , . . . , x n , x −1 1 , . . . , x −1 n . By using the Weyl character formula (2.12) and we have And we have First we consider the case where λ n ≥ 2. In this case, by the same argument as in the proof of (1), we have where λ runs over all partitions of length ≤ n.
Next we consider the case where λ n = 1. Then we have where c(ε, δ) = 1 if ε n = 0 and δ n = 1, 0 otherwise. Now, by the argument similar to that in the proof of (1), we obtain the desired result.

Applications to combinatorics of oscillating tableaux
In this section, we apply the Pieri rules obtained in the previous sections to derive several equinumeration results between down-up/up-down tableaux (generalization of oscillating tableaux) and column-strict/row-strict tableaux (generalization of standard tableaux).
Definition 5.1. A filling of the diagram of a partition λ with positive integers is called a column-strict (resp. row-strict) tableau if it satisfies the following two conditions: (i) Every row is weakly increasing (resp. strictly increasing), (ii) Every columns is strictly increasing (resp. weakly increasing).
Given a columns-strict or row-strict tableau T , the weight of T is defined to be the sequence (α 1 , α 2 , . . . ), where α i is the number of occurrences of i in T . We denote by CSTab(λ, α) (resp. RSTab(λ, α)) the set of all column-strict (resp. row-strict) tableaux of shape λ and weight α.
Proof. We consider the classical group G and its representations T and W listed in the following table: Here S α (V ) and α (V ) is defined by where V is the defining representation of G. We compute the "multiplicity" On the other hand, the defining representation of G is the restriction of the defining representation of GL N . Hence, by using the Pieri rule (2.1) for GL N , we have the following decomposition as GL N -modules: where CSTab(λ, α) (resp. RSTab(λ, α)) denotes the set of column-strict (resp. row-strict) tableaux of shape λ and weight α.  where µ runs over all partitions of length ≤ 2n.
It would be interesting to find bijective proofs of Theorem 5.3 and Corollary 5.5 by generalizing the arguments in [2] and [7].