Hall-Littlewood symmetric functions via Yamanouchi toppling game

TL;DR

This paper introduces the Yamanouchi toppling game on graphs, linking it to Young tableaux and Hall-Littlewood symmetric functions, providing new combinatorial insights into symmetric functions and orthogonal polynomials.

Contribution

It establishes a novel combinatorial framework connecting toppling games, Young tableaux, and symmetric functions, specifically relating the game to Hall-Littlewood polynomials.

Findings

Firing sequences correspond to standard Young tableaux.

The formal power series encodes configurations and reduces to Hall-Littlewood polynomials.

The approach offers a combinatorial perspective on orthogonal polynomials.

Abstract

We define a solitary game, the Yamanouchi toppling game, on any connected graph of n vertices. The game arises from the well-known chip-firing game when the usual relation of equivalence defined on the set of all configurations is replaced by a suitable partial order. The set all firing sequences of length m that the player is allowed to perform in the Yamanouchi toppling game is shown to be in bijection with all standard Young tableaux whose shape is a partition of the integer m with at most n-1 parts. The set of all configurations that a player can obtain from a starting configuration is encoded in a suitable formal power series. When the graph is the simple path and each monomial of the series is replaced by a suitable Schur polynomial, we prove that such a series reduces to Hall-Littlewod symmetric polynomials. The same series provides a combinatorial description of orthogonal polynomials when the monomials are replaced by products of moments suitably modified.