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arXiv · 1604.07214

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

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

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Yuki Irie. 2016-04-25. p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups. https://doi.org/10.1007/s10801-017-0799-6

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