Gorenstein properties and integer decomposition properties of lecture hall polytopes

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.