p-Saturations of Welter’s game and the irreducible representations of symmetric groups

Abstract

We establish a relation between the Sprague–Grundy function Open image in new window of p-saturations of Welter’s game and the degrees of the ordinary irreducible representations of symmetric groups. In these games, a position can be regarded as a partition \(\lambda \). Let \(\rho ^\lambda \) be the irreducible representation of the symmetric group \(\mathrm{Sym}(\left| \lambda \right| )\) corresponding to \(\lambda \). For every prime p, we show the following results: (1) \(\mathrm{sg}(\lambda ) \le \left| \lambda \right| \) with equality if and only if the degree of \(\rho ^\lambda \) is prime to p; (2) the restriction of \(\rho ^\lambda \) to \(\mathrm{Sym}(\mathrm{sg}(\lambda ))\) has an irreducible component with degree prime to p. Further, for every integer p greater than 1, we obtain an explicit formula for \(\mathrm{sg}(\lambda )\).