p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups

TL;DR

This paper links the Sprague-Grundy function of a p-saturation of Welter's game with the degrees of symmetric group representations, providing new formulas and characterizations for these mathematical objects.

Contribution

It establishes a novel relationship between game theory and representation theory, offering explicit formulas and criteria for degrees of irreducible representations related to Welter's game.

Findings

Degree of representation is prime to p iff the Sprague-Grundy value equals the size of the partition

Restriction of representation contains an irreducible component with degree prime to p

Explicit formula for the Sprague-Grundy function for any integer p > 1

Abstract

We establish a relation between the Sprague-Grundy function $\text{sg}$ of a $p$-saturation of Welter's game and the degrees of the ordinary irreducible representations of symmetric groups. In this game, a position can be viewed as a partition $\lambda$. Let $\rho^\lambda$ be the irreducible representation of $\text{Sym}(|\lambda|)$ indexed by $\lambda$. For every prime $p$, we show the following results: (1) the degree of $\rho^\lambda$ is prime to $p$ if and only if $\text{sg}(\lambda) = |\lambda|$; (2) the restriction of $\rho^\lambda$ to $\text{Sym}(\text{sg}(\lambda))$ has an irreducible component with degree prime to $p$. Further, for every integer $p$ greater than 1, we obtain an explicit formula for $\text{sg}(\lambda)$.