克罗内克系数与同时共轭类
Kronecker Coefficients and Simultaneous Conjugacy Classes
AI总结：
该研究引入克罗内克赫克代数研究有限群的克罗内克系数，建立同时共轭类数量与克罗内克系数的关系，定义双实群并刻画其与无重数张量积的关联，证明多类有限群不具有无重数张量积。
摘要：
克罗内克系数是有限群$G$的不可约表示在不可约表示张量积中的重数。我们定义了克罗内克赫克代数（Kronecker Hecke algebras），并将其作为研究有限群中克罗内克系数的工具。我们证明，有限群$G$中同时共轭类的数量等于克罗内克系数的平方和，而在元素逐元取逆下封闭的同时共轭类的数量等于以弗罗贝尼乌斯-舒尔指标（Frobenius-Schur indicators）加权的克罗内克系数之和。我们利用这些工具研究了哪些有限群具有无重数张量积。我们引入了双实群（doubly real groups）这一类群，并证明它们恰好是具有无重数张量积的实群。我们还证明，奇数阶非阿贝尔群、非阿贝尔有限单群以及大多数有限一般线性群都不具有无重数张量积。
英文摘要：
A Kronecker coefficient is the multiplicity of an irreducible representation of a finite group $G$ in a tensor product of irreducible representations. We define Kronecker Hecke algebras and use them as a tool to study Kronecker coefficients in finite groups. We show that the number of simultaneous conjugacy classes in a finite group $G$ is equal to the sum of squares of Kronecker coefficients, and the number of simultaneous conjugacy classes that are closed under elementwise inversion is the sum of Kronecker coefficients weighted by Frobenius-Schur indicators. We use these tools to investigate which finite groups have multiplicity-free tensor products. We introduce the class of doubly real groups, and show that they are precisely the real groups which have multiplicity-free tensor products. We show that non-Abelian groups of odd order, non-Abelian finite simple groups, and most finite general linear groups do not have multiplicity-free tensor products.
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