Hibi rings and the Ehrhart rings of chain polytopes - Part 1

Abstract: In 1985, Stanley submitted a paper titled "Two Poset Polytopes", which was published in 1986, in which he defined the order and chain polytopes of a finite partially ordered set (poset for short). On the other hand, Hibi presented a notion of an algebra with straightening law (ASL for short) on a finite distributive lattice, which nowadays is called a Hibi ring, in a conference held in Kyoto 1985. This result was published in 1987. It turned out that the Hibi ring on a distributive lattice D is the Ehrhart ring of the order polytope of the poset consisting of join-irreducible elements of D. In the first talk, we recall the definition of Ehrhart rings, order and chain polytopes, and Hibi rings. We recall some basic properties of Ehrhart rings and describe the canonical module of them. Using these facts, we state some basic facts of Hibi rings, i.e., the Ehrhart rings of the order polytopes of posets. We also state some basic facts of the Ehrhart rings of chain polytopes of posets. In the second talk, we focus on the structure of the canonical modules of the Ehrhart rings of order and chain polytopes of a poset. We describe the generators of the canonical modules in terms of the combinatorial structure of the poset and characterize the level property. If time permits, we describe the radical of the trace of the canonical module of these rings and describe the non-Gorenstein locus. This final part is a joint-work with Janet Page.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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