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arXiv · 1912.12573

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

Yuki Irie
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Abstract

For impartial games $\Gamma$ and $\Gamma'$, the Sprague-Grundy function of the disjunctive sum $\Gamma + \Gamma'$ 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 $\Gamma'$, the Sprague-Grundy function of a $p$-saturation of $\Gamma + \Gamma'$ 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.

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Yuki Irie. 2019-12-29. A base-p Sprague-Grundy type theorem for p-calm subtraction games: Welter's game and representations of generalized symmetric groups. https://arxiv.org/abs/1912.12573

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