Hibi Algebras and Representation Theory

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 [2,5,6,10,33]. In particular, a special type of such algebras introduced by Hibi [12] 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.2010 Mathematics Subject Classification. 13A50, 13F50, 20G05, 05E10, 05E15.

Toric non-Gorenstein loci

“…Nowadays Hibi rings are objects of interest in their own, and appear in various geometric, algebraic and combinatorial contexts. Consider for example [HH05], [EHM11], [How05], [KP18]. The non-Gorenstein loci of Hibi rings have been subject to extensive study: a Hibi ring k[P ] is Gorenstein if and only if P is a pure poset, that is, all maximal chains of P have the same length ([Hib87], Corollary 3.d).…”

On the regularity of join-meet ideals of modular lattices

“…In literature, the ring R[L] is known as the Hibi ring of the distributive lattice L. Hibi rings and the corresponding varieties appear naturally in different combinatorial, algebraic, and geometric contexts; see [3,4,10,11,14,15,20]. Paper [12] surveys the relation between Hibi rings and representation theory.…”