Character tables and defect groups

2020/07/09 by Sambale, Benjamin
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2007.04919

Abstract

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D'|=4 where D' denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.

Related

• Real characters in nilpotent blocks
• Frobenius-Schur indicators of characters in blocks with cyclic defect
• Character degrees in blocks and defect groups
• On character degrees of cyclic and Klein four defect blocks
• Notes on Loewy series of centers of p-blocks