Self-dual modules in characteristic two and normal subgroups

JOURNAL OF ALGEBRA

卷:570

Article

Gow, Rod¹  Murray, John² 

[1] Univ Coll Dublin, Sch Math Sci, Dublin, Ireland

[2] Maynooth Univ, Dept Math & Stat, Maynooth, Kildare, Ireland

关键词: Finite groups;    Representation theory;    Brauer characters;    Clifford theory;    Duality;    Quadratic modules;    Brauer blocks;   

DOI  :  10.1016/j.jalgebra.2020.11.014

来源: Elsevier

【 摘 要 】

We prove Clifford theoretic results which only hold in characteristic 2. Let G be a finite group, let N be a normal subgroup of G and let phi be an irreducible 2-Brauer character of N. We show that phi occurs with odd multiplicity in the restriction of some self-dual irreducible Brauer character theta of G if and only if phi is G-conjugate to its dual. Moreover, if is self-dual then theta is unique and the multiplicity is 1. Next suppose that theta is a self-dual irreducible 2-Brauer character of G which is not of quadratic type. We prove that the restriction of theta to N is a sum of distinct self-dual irreducible Brauer character of N, none of which have quadratic type. Moreover, G has no self-dual irreducible 2-Brauer character of non-quadratic type if and only if N and GIN satisfy the same property. Finally, suppose that b is a real 2-block of N. We show that there is a unique real 2-block of G covering b which is weakly regular with respect to N. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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