Distributive lattices, affine semigroups, and branching rules of the classical groups

Abstract

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.2000 Mathematics Subject Classification. 20G05, 05E15.

Tiling Branching Multiplicity Spaces With Gl pattern Blocks

“…Hom GL 2 (V (5,1) 2 , V (8,4,2,0) 4 ), Hom GL 2 (V (5,2) 2 , V (8,4,1,0) 2 ) S. Kim [11] which are, by Theorem 3.5, as GL 2 representations, isomorphic to C ⊗ V (8,5) 2…”

“…We can also obtain similar results for the orthogonal group within certain stable ranges. For more about stable range conditions in branching rules for classical groups, we refer readers to [4]. 5.1.…”

Double Pieri algebras and iterated Pieri algebras for the classical groups

“…This partial order is the opposite of the standard partial order on the set of column tableaux ( [HL2,Ki4]). Let T ∈ ST n,k, .…”

“…Recently, Kim has shown that the branching algebra from Sp 2n to Sp 2k is a deformation of a Hibi ring, and in fact, isomorphic in certain ranges to an associated branching algebra for GL 2n . He has also shown that analogous branching algebras for orthogonal groups, from SO n to SO k , subject to a suitable stability condition, are deformations of Hibi rings (see [Ki2,Ki4,KY]). …”

Hibi Algebras and Representation Theory

“…Then, the multichains of B n,m,k , the corresponding GT patterns, and the Hibi algebra attached to them can be used to describe branching rules for some pairs (G, H) of classical groups, that is, how a representation of G decomposes into irreducible representations of a subgroup H of G. See [23,26,31,32].…”