Welter对策的p-饱和与对称群的不可约表示
p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups
摘要：
建立了Welter对策的$p$ -饱和的Sprague-Grundy函数$\text{sg}$与对称群的普通不可约表示的度之间的关系.在这个游戏中，一个位置可以被看作是一个分区$λ$。令$ρ^λ$是由$λ$索引的$\text{Sym}(|λ|)$的不可约表示。对每一个素数$p$，我们证明了以下结果：（1）$ρ^λ$的度与$p$互素当且仅当$\text{sg}(λ) = |λ|$ ;（2）$ρ^λ$对$\text{Sym}(\text{sg}(λ))$的限制有一个度与$p$互素的不可约分支.进一步地，对于每个大于1的整数$p$，我们得到了一个关于$\text{sg}(λ)$的显式公式.
英文摘要：
We establish a relation between the Sprague-Grundy function $\text{sg}$ of a $p$-saturation of Welter's game and the degrees of the ordinary irreducible representations of symmetric groups. In this game, a position can be viewed as a partition $λ$. Let $ρ^λ$ be the irreducible representation of $\text{Sym}(|λ|)$ indexed by $λ$. For every prime $p$, we show the following results: (1) the degree of $ρ^λ$ is prime to $p$ if and only if $\text{sg}(λ) = |λ|$; (2) the restriction of $ρ^λ$ to $\text{Sym}(\text{sg}(λ))$ has an irreducible component with degree prime to $p$. Further, for every integer $p$ greater than 1, we obtain an explicit formula for $\text{sg}(λ)$.
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