Séminaire Lotharingien de Combinatoire, 84B.25 (2020), 12 pp.

Petter Brändén and Liam Solus

Some Algebraic Properties of Lecture Hall Polytopes

Abstract. In this note, we investigate some of the fundamental algebraic and geometric properties of s-lecture hall simplices and their generalizations. We show that all s-lecture hall order polytopes, which simultaneously generalize s-lecture hall simplices and order polytopes, satisfy a property which implies the integer decomposition property. This answers one conjecture of Hibi, Olsen and Tsuchiya. By relating s-lecture hall polytopes to alcoved polytopes, we then use this property to show that families of s-lecture hall simplices admit a quadratic Gröbner basis with a square-free initial ideal. Consequently, we find that all <s-lecture hall simplices for which the first order difference sequence of s is a 0,1-sequence have a regular and unimodular triangulation. This answers a second conjecture of Hibi, Olsen and Tsuchiya, and it gives a partial answer to a conjecture of Beck, Braun, Köppe, Savage and Zafeirakopoulos.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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