A base-\(p\) Sprague-Grundy type theorem for \(p\)-calm subtraction games: Welter’s game and representations of generalized symmetric groups. (English) Zbl 1490.91039
Summary: 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 ap-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.
MSC:
| 91A46 | Combinatorial games |
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