Gorenstein properties and integer decomposition properties of lecture hall polytopes

Takayuki Hibi^a, McCabe Olsen^b, Akiyoshi Tsuchiya<sup>a

a</sup> Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
^b Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

Abstract: Though much is known about $\mathbf{s}$-lecture hall polytopes, there are still many unanswered questions. In this paper, we show that $\mathbf{s}$-lecture hall polytopes satisfy the integer decomposition property (IDP) in the case of monotonic $\mathbf{s}$-sequences. Given restrictions on a monotonic $\mathbf{s}$-sequence, we discuss necessary and sufficient conditions for the Fano, reflexive and Gorenstein properties. Additionally, we give a construction for producing Gorenstein/IDP lecture hall polytopes.

Key words and phrases: lecture hall polytopes, Gorenstein polytopes, integer decomposition property.

MSC: 05A20, 05E40, 13P20, 52B20

Language: English

DOI: 10.17323/1609-4514-2018-18-4-667-679

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