p-Saturations of Welter’s game and the irreducible representations of symmetric groups
p-Saturations of Welter’s game and the irreducible representations of symmetric groups
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Welter 博弈的 p-饱和度和对称群的不可约表示
DOI：
10.1007/s10801-017-0799-6
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发表时间：
2016
影响因子：
0.8
通讯作者：
Yuki Irie
中科院分区：
文献类型：
--
作者：
Yuki Irie
我们在这些游戏中的Sprague-Grundy函数与普通的不可减至的表达式之间的关系之间建立了关系$$ \ rho ^\ lambda $$ρλ是对称组的不可约代表，$$ \ mathrm {sym}（\ left | \ lambda \ right |）$ sym sym（λ）每一个prime，我们显示以下结果：（1）$$ \ mathrm {sg}（\ lambda）\ le \ left | \ lambda \ right | ^\ lambda $$ρλ是p的; SG（λ）具有不可修复的成分，prime tem prime to p prime。 $$ \ mathrm {sg}（\ lambda）$$ sg（λ）的公式。
We establish a relation between the Sprague–Grundy function of p-saturations of Welter’s game and the degrees of the ordinary irreducible representations of symmetric groups. In these games, a position can be regarded as a partition $$\lambda $$λ. Let $$\rho ^\lambda $$ρλ be the irreducible representation of the symmetric group $$\mathrm{Sym}(\left| \lambda \right| )$$Sym(λ) corresponding to $$\lambda $$λ. For every prime p, we show the following results: (1) $$\mathrm{sg}(\lambda ) \le \left| \lambda \right| $$sg(λ)≤λ with equality if and only if the degree of $$\rho ^\lambda $$ρλ is prime to p; (2) the restriction of $$\rho ^\lambda $$ρλ to $$\mathrm{Sym}(\mathrm{sg}(\lambda ))$$Sym(sg(λ)) has an irreducible component with degree prime to p. Further, for every integer p greater than 1, we obtain an explicit formula for $$\mathrm{sg}(\lambda )$$sg(λ).