Hibi algebras and representation theory

TL;DR

This paper surveys how Hibi algebras connect combinatorics and representation theory, especially through poset structures, affine monoids, and toric degenerations of flag varieties, enhancing understanding of classical group representations.

Contribution

It highlights the role of Hibi algebras in linking combinatorial structures with the representation theory of classical groups, emphasizing their applications in algebraic geometry and invariant theory.

Findings

Hibi algebras serve as a bridge between combinatorics and representation theory.

Poset structures of Young tableaux relate to Hibi algebras in flag varieties.

Hibi algebras facilitate toric degenerations of algebraic varieties.

Abstract

This paper gives a survey on the relation between Hibi algebras and representation theory. The notion of Hodge algebras or algebras with straightening laws has been proved to be very useful to describe the structure of many important algebras in classical invariant theory and representation theory. In particular, a special type of such algebras introduced by Hibi provides a nice bridge between combinatorics and representation theory of classical groups. We will examine certain poset structures of Young tableaux and affine monoids, Hibi algebras in toric degenerations of flag varieties, and their relations to polynomial representations of the complex general linear group.