A base-p Sprague-Grundy type theorem for p-calm subtraction games: Welter's game and representations of generalized symmetric groups

Abstract

For impartial games $\Gamma$ and ${\Gamma }^{\prime }$, the Sprague-Grundy function of the disjunctive sum $\Gamma +{\Gamma }^{\prime }$ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce $p$-calm subtraction games, and show that for $p$-calm subtraction games $\Gamma$ and ${\Gamma }^{\prime }$, the Sprague-Grundy function of a $p$-saturation of $\Gamma +{\Gamma }^{\prime }$ is equal to the $p$-Nim-sum of the Sprague-Grundy functions of their $p$-saturations. Here a $p$-Nim-sum is the result of addition without carrying in base $p$ and a $p$-saturation of $\Gamma$ is an impartial game obtained from $\Gamma$ by adding some moves. It will turn out that Nim and Welter's game are $p$-calm. Further, using the $p$-calmness of Welter's game, we generalize a relation between Welter's game and representations of symmetric groups to disjunctive sums of Welter's games and representations of generalized symmetric groups; this result is described combinatorially in terms of Young diagrams.