The Lecture Hall Cone as a toric deformation

The Lecture Hall cone is a simplicial cone whose lattice points naturally correspond to Lecture Hall partitions. The celebrated Lecture Hall Theorem of Bousquet-M\'elou and Eriksson states that a particular specialization of its multivariate Ehrhart series factors in a very nice and unexpected way. Over the years, several proofs of this result have been found, but it is still not considered to be well-understood from a geometric perspective. In this note we propose two conjectures which aim at clarifying this result. Our main conjecture is that the Ehrhart ring of the Lecture Hall cone is actually an initial subalgebra $A_n$ of a certain subalgebra of a polynomial ring, which is itself isomorphic to a polynomial ring. As passing to initial subalgebras does not affect the Hilbert function, this explains the observed factorization. We give a recursive definition of certain Laurent polynomials, which generate the algebra $A_n$. Our second conjecture is that these Laurent polynomials are in fact polynomials. We computationally verified that both conjectures hold for Lecture Hall partitions of length at most 12.