Unimodular Triangulations and Ehrhart Non-Positivity for s-Lecture Hall Simplices

For a sequence of positive integers $s = (s_1, ..., s_n)$, the s-lecture hall simplex is defined as $P_n^s = Conv{ (0, ..., 0), (0, ..., 0, s_n), ..., (s_1, ..., s_n)}$. Hibi, Olsen, and Tsuchiya (2016) conjectured that for any sequence $s$, the simplex $P_n^s$ admits a unimodular triangulation (UT). Partial results toward this conjecture were obtained by Brändén and Solus in 2019, and by Hibi, Olsen, and Tsuchiya in 2016.

In this work, we provide further evidence for this conjecture by extending previous results. Specifically, we show that if $P_n^s$ admits a UT, then so does $P_n^{s'}$, where $s' = (s_1, ..., s_n + 1)$. Moreover, we study Ehrhart-theoretic aspects of certain lecture hall simplices. Our main result states that for $s = (a, ..., a, a + 1)$ with $a > 0$ and $n > 4$, the simplex $P_n^s$ is not Ehrhart positive for $a$ large enough.

This is joint work with Martina Juhnke and Germain Poullot.