On the conjugating representation of a finite group
On the conjugating representation of a finite group
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关于有限群的共轭表示
DOI：
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
K. Fields
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文献类型：
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作者：
K. Fields
人们对有限群的共轭表示如何分解为不可约表示知之甚少。在这篇文章中，我们研究了在这种分解中哪些重数集是可能的。常规代表的类似问题也长期悬而未决。令vG = C • 1G + yG 表示G（或其字符）的共轭表示。如果c表示G的共轭类数，则主表示1G不会出现在yG的分解中。我们在这里证明，如果 G 不是交换矩阵（即，如果 7G ^ 0），则 yG 包含至少两个不等价的不可约表示；此外，如果群具有平凡中心且不是三个字母上的对称群，则 yG 不是无重数的；如果群很简单，那么 g.c.d. yG 的不可约成分的次数不能被 G 的任何不可约表示的次数整除。回想一下，主要表示是单个不可约表示的副本的直接和。
Very little is known about how the conjugating representation of a finite group decomposes into irreducible representations. In this note we investigate which sets of multiplicities are possible in such a decomposition. The analogous question for the regular representation is also long unsolved. Let vG = C • 1G + yG denote the conjugating representation of G (or its character). If c denotes the number of conjugacy classes of G, then the principal representation 1G does not appear in the decomposition of yG. We show here that if G is not abelian (i.e., if 7G ^ 0) then yG contains at least two inequivalent irreducible representations; moreover, if the group has trivial center and is not the symmetric group on three letters, then yG is not multiplicity-free; and if the group is simple, then the g.c.d. of the degrees of the irreducible constituents of yG is not divisible by the degree of any irreducible representation of G. Recall that a primary representation is a direct sum of copies of a single irreducible representation.