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p -Blocks Relative to a Character of a Normal Subgroup II
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-10-13 , DOI: 10.1007/s00009-020-01599-z
Noelia Rizo

Let p be a prime number, let G be a finite group, let N be a normal subgroup of G, and let \(\theta \) be a G-invariant irreducible character of N. In Rizo (J Algebra 514:254–272, 2018), we introduced a canonical partition of the set \(\mathrm{Irr}(G|\theta )\) of irreducible constituents of the induced character \(\theta ^G\), relative to the prime p. We call the elements of this partition the \(\theta \)-blocks. In this paper, we construct a canonical basis of the complex space of class functions defined on \(\{ x \in G \, |\, x_p \in N\}\), which supersedes previous non-canonical constructions. This allows us to define \(\theta \)-decomposition numbers in a natural way. We also prove that the elements of the partition of \({\text {Irr}}(G|\theta )\) established by these \(\theta \)-decomposition numbers are the \(\theta \)-blocks.



中文翻译:

相对于正常子群II的一个特征的p块

p为素数,令G为有限群,令NG的正规子群,令\(\ theta \)NG不变不可约性。在Rizo(J Algebra 514:254–272,2018)中,我们引入了诱导字符\(\ theta ^ G的不可约成分的集合\(\ mathrm {Irr}(G | \ theta)\)的规范划分\)相对于素数p。我们将此分区的元素称为\(\ theta \)- blocks。在本文中,我们构造了\(\ {x \ in G \,| \,x_p \ in N \} \)上定义的类函数的复杂空间的规范基础。,它取代了以前的非规范构造。这使我们可以自然地定义\(\ theta \)-分解数字。我们还证明了由这些\(\ theta \)分解数字建立的\({\ text {Irr}}(G | \ theta)\)分区的元素是\(\ theta \)-块

更新日期:2020-10-13
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