City Research Online

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 nonprincipal 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 of 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.

Publication Type:	Article
Additional Information:	This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics
SWORD Depositor:	Symplectic Administrator

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