对称群的Character值与分解矩阵
Character values and decomposition matrices of symmetric groups
摘要：
利用对称群的普通特征标取值之间的关系，证明了对称群的模表示理论中的两个定理。1.具有奇特征的对称群的分解矩阵具有不同的行。在特征2中，由不同分区$λ$和$μ$标记的分解矩阵的行是相等的，当且仅当$λ$和$μ$是共轭的。对Hecke代数证明了一个类似的结果。2.定义在奇特征的代数闭域上的对称群$S_n$的Specht模在交错群$A_n$的限制下是可分解的当且仅当它是单的，并且标号划分是自共轭的。这个结果被推广到任意域的奇特性。
英文摘要：
By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions $λ$ and $μ$ are equal if and only if $λ$ and $μ$ are conjugate. An analogous result is proved for Hecke algebras. 2. A Specht module for the symmetric group $S_n$, defined over an algebraically closed field of odd characteristic, is decomposable on restriction to the alternating group $A_n$ if and only if it is simple, and the labelling partition is self-conjugate. This result is generalised to an arbitrary field of odd characteristic.
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