Distributive Lattices, Affine Semigroups, and Branching Rules of the Classical Groups

TL;DR

This paper explores the algebraic structures underlying branching rules of classical groups, using distributive lattices and affine semigroups to describe stable range branching algebras via toric degenerations.

Contribution

It introduces a unified combinatorial framework using distributive lattices and Hibi algebras to describe stable range branching algebras for classical groups.

Findings

Identification of distributive lattices for stable branching rules

Construction of Hibi algebras representing these branching algebras

Connection to toric degenerations of spherical varieties

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.