Structure of blocks with normal defect and abelian p' inertial quotient

2022/01/27 by David J. Benson, David Benson, Benson, David +4
Mathematics · #20C20 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #Rings, Modules, and Algebras #math.RT #msc:20C20

paper · pdf · doi:10.48550/arxiv.2201.11715

21 pages

arxiv created 2022/01/27 · openalex publication_date 2022/01/27 · arxiv updated 2022/01/28 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

Let k be an algebraically closed field of prime characteristic p. Let kGe be a block of a group algebra of a finite group G, with normal defect group P and abelian p' inertial quotient L. Then we show that kGe is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings--Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the non-principal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order p³ with a quaternion group of order eight with the centre acting trivially. In the case p=3 we give explicit generators and relations for the basic algebra as a quantised version of kP. As a second example, we give explicit generators and relations in the case of a group of shape 2¹⁺⁴:3¹⁺² in characteristic two.

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