Counting blocks of defect zero
Counting blocks of defect zero
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DOI：
10.1007/978-3-0348-8658-1_18
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
R. Knörr
中科院分区：
文献类型：
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作者：
R. Knörr
求一个有限群和一个素数。那么缺陷0 g的p块数，即g与的一般不可约特征的个数，显然是群代数f g中心的合适理想的维数，其中f是一个特征域。众所周知，如果g有一个非平凡的正规子群q，这个数是零。然而，在这种情况下，人们仍然可以要求缺陷为零的因子群的块数，即gwithq≤kerχ的普通不可约字符的数量，这将再次证明这是z （FG）中合适的理想（将在下面描述）的维数。事实上，可以稍微精确一点：给定块b1，…，BnofG，可以将缺陷0 0中属于给定块之一的不可约字符作为g的字符进行计数。
LetGbe a finite group andpa prime number. Then the number ofp-blocks of defect zero inG — i.e.the number of ordinary irreducible charactersχofGwith- is clearly the dimension of a suitable ideal in the center of the group algebraFG, whereFis a field of characteristicp. It is well known that this number is zero, ifGhas a non-trivial normalp-subgroupQ, say. In this situation, however, one may still ask for the number of blocks of the factor groupwith defect zero -i.e.the number of ordinary irreducible characters χ ofGwithQ ≤KerχandIt will turn out that this again is the dimension of a suitable ideal (to be described below) inZ(FG). In fact, one can be slightly more precise: givenp-blocksB1,...,BnofG, one can count the irreducible characters of defect zero inGwhich belong to one of the given blocks when viewed as characters ofG.