A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem

TL;DR

This paper introduces a new combinatorial formula for affine Hall-Littlewood functions of type _{n-1}, representing them as weighted sums over lattice points in an infinite-dimensional polyhedron, using a generalized Brion's theorem.

Contribution

It provides the first combinatorial formula for affine Hall-Littlewood functions via a weighted Brion theorem approach, connecting representation theory and polyhedral geometry.

Findings

Derived a weighted version of Brion's theorem.

Expressed affine Hall-Littlewood functions as sums over lattice points.

Established a new combinatorial framework for affine symmetric functions.

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 $\hat{\mathfrak{sl}}_n$. 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.