Paths and Kostka--Macdonald Polynomials

TL;DR

This paper explores combinatorial models related to box-ball systems, connecting them with representation theory and providing new formulas and statistics that generalize existing energy measures.

Contribution

It offers multiple combinatorial descriptions of the state space, links partition functions to q-multiplicities, and introduces generalized energy statistics.

Findings

Connected combinatorial descriptions with representation theory

Provided an elementary proof of a special case of a known formula

Proposed new combinatorial statistics generalizing energy

Abstract

We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t=1 of the Haglund--Haiman--Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics.