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

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 $\text{sg}(\lambda )$.

24 pages