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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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