J Algebr Comb (2012) 35:269–290 DOI 10.1007/s10801-011-0303-7 A basis for the symplectic group branching algebra Sangjib Kim · Oded Yacobi Received: 13 May 2010 / Accepted: 22 June 2011 / Published online: 23 July 2011 © Springer Science+Business Media, LLC 2011 Abstract The symplectic group branching algebra, B, is a graded algebra whose C components encode the multiplicities of irreducible representations of Sp2n−2( ) in C B each finite-dimensional irreducible representation of Sp2n( ). By describing on an ASL structure, we construct an explicit standard monomial basis of B consisting of C B Sp2n−2( ) highest weight vectors. Moreover, is known to carry a canonical ac- tion of the n-fold product SL2 ×···×SL2, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Fi- nally, using the theory of Hibi algebras we describe a deformation of Spec(B) into an explicitly described toric variety. Keywords Symplectic groups · Branching rules · Hibi algebra · Algebra with straightening law 1 Introduction Let us consider a pair of complex reductive algebraic groups G and H with embed- ding H ⊂ G, and irreducible representations VG and VH of G and H , respectively. A description of the multiplicity of VH in VG regarded as a representation of H by restriction is called a branching rule for (G, H ). By Schur’s lemma, the (branching) multiplicity space,HomH (VH ,VG), encodes the branching rule. S. Kim () School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia e-mail: [email protected] O. Yacobi Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S-2E4, Canada e-mail: [email protected] 270 J Algebr Comb (2012) 35:269–290 In this paper, we study the branching multiplicity spaces for the symplectic group C C − Sp2n( ) of rank n down to the symplectic group Sp2n−2( ) of rank n 1 by investi- gating the associated branching algebra. The branching algebra is a subalgebra of the C ring of regular functions over Sp2n( ). Moreover, it is graded, having the branching multiplicity spaces as its graded components: B =∼ D F HomSpn−1 τn−1,τn . (D,F )∈Λn−1,n C C Here D and F run over highest weights for Sp2n−2( ) and Sp2n( ), respectively, D F and τ2n−2 and τn are the associated finite-dimensional irreducible representations. C C Branching rules for (Sp2n( ), Sp2n−2( )), especially their combinatorial aspects, are well known (see, e.g., [8, 9, 11–13, 18]). The main goal of this paper is to investi- gate the branching algebra B which governs the branching of symplectic groups. Our main results are several-fold. Firstly, we describe on B an algebra with straightening law (ASL) structure which presents B simply in terms of generators and relations. Secondly, we show that this ASL structure is compatible with a canonical weight basis of B coming from a “hidden symmetry” of B, namely an action of the n-fold product of SL2 which acts irreducibly on the graded components of B. Finally, we unify these algebraic results with previously known combinatorial rules governing the branching of symplectic groups via a toric degeneration of B into an explicitly described toric variety. In Sect. 2, we construct the branching algebra B and review branching rules for C C B (Sp2n( ), Sp2n−2( )). Then in Sect. 3, we study from the perspective of an ASL over a distributive lattice. Our first result shows that B has a natural standard mono- mial basis which satisfies simple straightening relations. As a corollary we obtain a finite presentation of B in terms of generators and relations. Then, in Sect. 4, we recall a theorem from [21] which shows that the natural SL2 action on B can be canonically extended to an action of L = SL2 ×···×SL2 (n copies) C D F in such a way that each multiplicity space HomSp2n−2( )(τn−1,τn ) is an irreducible L-module. In particular, this theorem describes a canonical decomposition of B into one-dimensional spaces. Our second result shows this decomposition is compatible with the ASL structure on B. In other words, the standard monomial basis is the unique (up to scalar) weight basis for the action of L on B. In Sect. 5, we show that B can be flatly deformed into a Hibi algebra, and, as a corollary, that Spec(B) is a deformation of an explicitly described toric variety. In particular, this connects our enumeration of standard monomials with the more common description of branching rules using diagrams of interlacing weights. In future work we will apply these results to study properties of the canonical weight basis for irreducible representations of the symplectic group arising from this work. J Algebr Comb (2012) 35:269–290 271 C C 2 Branching algebra for (Sp2n( ), Sp2n−2( )) In this section we introduce our main object of study, the branching algebra for the C C pair (Sp2n( ), Sp2n−2( )). 2.1 Symplectic groups Let Qn = (qa,b) be the n×n matrix with qa,n+1−a = 1for1≤ a ≤ n and 0 otherwise. Then we define the symplectic group 2n Gn = Sp C ,Qn of rank n as the subgroup of the general linear group GL(2n, C) preserving the skew symmetric bilinear form on C2n induced by 0 Qn . −Qn 0 2n Let {ea} be the elementary basis of C , and denote by {ea,ea} the isotropic pairs, where ea = e2n+1−a for 1 ≤ a ≤ n. With respect to this bilinear form, we can take the subgroup Un of Gn consisting of upper triangular matrices with 1’s on the diagonal − as a maximal unipotent subgroup of Gn.WeletUn denote the subgroup of lower triangular matrices with 1’s on the diagonal. Let us identify Gn−1 with the subgroup of Gn which acts as identity on the isotropic space spanned by {en,en }. Then Gn−1 can be embedded in Gn via ⎡ ⎤ A 0 B AB ⎢ ⎥ → ⎣ 0 I 0 ⎦ (2.1) CD C 0 D where A,B,C,D are (n − 1) × (n − 1) matrices, I is the 2 × 2 identity matrix, and 0’s are the zero matrices of proper sizes. A Young diagram is a finite left-justified array of boxes with weakly decreasing row lengths, such as We shall identify a Young diagram F with its sequence of row lengths (f1,f2,...). By reading column lengths of F , we obtain its associated Young diagram F t called the transpose of F . We write (F ) for the number of non-zero entries in F and call it the length of F . The Young diagram in the above example is (6, 4, 2, 1) or equivalently (6, 4, 2, 1, 0,...)and (F ) = 4. Its transpose F t is (4, 3, 2, 2, 1, 1). 272 J Algebr Comb (2012) 35:269–290 Recall that every finite-dimensional irreducible representation of Gn can be uniquely labeled by a Young diagram with less than or equal to n rows by identi- fying its highest weight μF with Young diagram F = (f1,...,fn): = f1 ··· fn μF (t) t1 tn . Here t is an element of the maximal torus Tn of Gn = −1 −1 Tn diag t1,...,tn,tn ,...,t1 ∈ Zn ≥···≥ ≥ F and F with f1 fn 0. See [5, Sect. 3.2.1] for details. We let τn denote the irreducible representation of Gn labeled by Young diagram F . D F To encode the branching multiplicities of τn−1 in τn for all pairs of Young dia- grams (D, F ), we will use the semigroup = ∈ Zn−1 × Zn : ≤ − ≤ Λn−1,n (D, F ) ≥0 ≥0 (D) n 1,(F) n and construct an algebra graded by Λn−1,n. The semigroup structure of Λn−1,n is Zn−1 × Zn induced by the natural embedding of Λn−1,n in ≥0 ≥0 with addition. 2.2 Branching algebra B On the ring R(Gn) of regular functions over Gn, we have the natural action of Gn × Gn given by · = −1 (g1,g2) f (x) f g1 xg2 (2.2) for f ∈ R(Gn) and (g1,g2) ∈ Gn × Gn. With respect to this action, let us consider − × the affine quotient of Gn by Un 1. Lemma 2.1 [5, Theorem 12.1.5] As a Gn-module under right translation the − × R (Un 1)-invariant subalgebra of (Gn) contains every irreducible rational rep- resentation of Gn with multiplicity one: −× R Un 1 = F (Gn) τn . (2.3) F ∈Λn U −×1 The algebra R(Gn) n is graded by the semigroup of dominant weights for Gn or equivalently the set Λn of Young diagrams of length less than or equal to n. −× F R Un 1 In this setting, the irreducible representation τn is the weight space of (Gn) under the left action of the maximal torus Tn with weight μ(−F), i.e., − − · = f1 ··· fn t f t1 tn f ∈ F ∈ for f τn and t Tn. See [5, Sect. 12.1.3] for further details. F By highest weight theory (e.g., [5, Sect. 3.2.1]) the subspace of τn invariant under the maximal unipotent subgroup of Gn−1 is spanned by highest weight vectors of J Algebr Comb (2012) 35:269–290 273 F Gn−1-irreducible representations in τn . Therefore, the Un−1-invariant subalgebra of U −×1 R(Gn) n contains the information of the branching multiplicities for (Gn,Gn−1). That is, − × R Proposition 2.2 The (Un Un−1)-invariant subalgebra of the ring (Gn) decom- poses as −× U − R Un Un−1 = D F ⊗ D n 1 (Gn) HomGn−1 τn−1,τn τn−1 .