A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

Abstract

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type $${{\tilde{A}}}_{n-1}$$ , i.e., corresponding to the affine Lie algebra $${{\widehat{\mathfrak {sl}}}}_n$$ . Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula.