Vertices for irreducible characters of a class of blocks.
type: Item
instance of: scholarly article
title: Vertices for irreducible characters of a class of blocks.
zbMATH Open document ID: 1069.20007
author: Charles W. Eaton
published in: Journal of Algebra
publication date: 2005-05-04T00:00:00Z
full work available at URL: verticesarticle.pdf
review text: Let \(p\) be a prime, and let \(B\) be a \(p\)-block of a finite group \(G\) whose defect groups are contained in a \(p\)-solvable normal subgroup \(N\) of \(G\). Extending an idea of \textit{G. Navarro} [Trans. Am. Math. Soc. 354, No. 7, 2759-2773 (2002; Zbl 0998.20012)] the author associates with every irreducible character \(\chi\) in \(B\) pairs \((Q,\delta)\) consisting of a \(p\)-subgroup \(Q\) of \(G\) and an irreducible character \(\delta\) of \(Q\). These pairs are called the vertices of \(\chi\); they form a \(G\)-orbit under conjugation and are independent of the choice of the \(p\)-solvable normal subgroup \(N\). If \((Q,\delta)\) is a vertex of \(\chi\) then \(\chi\) is afforded by a relatively \(Q\)-projective module; in particular, \(\chi\) vanishes on elements whose \(p\)-factor is not conjugate to an element in \(Q\). Moreover, the defect of \(\chi\) coincides with the defect of \(\delta\), and \(Q=O_p(N_G(Q,\delta))\). Also, the irreducible characters in \(B\) with vertex \((Q,1_Q)\) are in canonical bijection with the irreducible Brauer characters of \(G\) with vertex \(Q\) (in Green's sense).
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 2164793
zbMATH Keywords: irreducible characters
zbMATH Keywords: blocks
zbMATH Keywords: defect groups
zbMATH Keywords: vertices
zbMATH Keywords: nuclei
zbMATH Keywords: Brauer characters
MaRDI profile type: Publication
OpenAlex ID: W2034516948
cites work: The Green correspondence and normal subgroups
cites work: Q3722697
cites work: Vertices of Simple Modules over p -Solvable Groups
cites work: Q3922829
cites work: Q3955560
cites work: A Note on Lifting Brauer Characters
cites work: Characters of \(\pi\)-separable groups
cites work: Characters of Relatively Projective Modules II
cites work: Vertices for characters of \(p\)-solvable groups
cites work: Defect of characters and local subgroups in p-solvable groups
cites work: On Characters of Relatively Projective Modules
DOI: 10.1016/J.JALGEBRA.2005.01.007
Recommended article: A note on the number of irreducible characters in a p-block with normal defect group
Recommended article: Characters, exponents and defects in \(p\)-solvable groups
Recommended article: Brauer characters and the Harris-Knörr correspondence in \(p\)-solvable groups.
Recommended article: A remark on covering blocks
Recommended article: Invariant constituents and invariant blocks under coprime action
Recommended article: On characters with minimal defects.
Recommended article: Vertices of simple modules and normal subgroups of \(p\)-solvable groups
Recommended article: Character degrees and blocks of finite groups
Recommended article: Defect of characters and local subgroups in p-solvable groups
Recommended article: On subsections of blocks and Brauer pairs
description: scientific article; zbMATH DE number 2164793