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

الأصل · EN

We establish a relation between the Sprague-Grundy function 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 λ. Let ρλ be the irreducible representation of Sym(|λ|) indexed by λ. For every prime p, we show the following results: (1) the degree of ρλ is prime to p if and only if sg(λ) = |λ|; (2) the restriction of ρλ to Sym(sg(λ)) has an irreducible component with degree prime to p. Further, for every integer p greater than 1, we obtain an explicit formula for sg(λ).