BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

Abstract

Let k be an algebraically closed field of characteristic p, and let O be either k or its ring of Witt vectors W (k). Let G a finite group and B a block of OG with normal abelian defect group and abelian p ′ inertial quotient. We show that B is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan's conjecture. For O = k, we give an explicit description of the basic algebra of B as a quiver with relations. It is a quantised version of the group algebra of the semidirect product P ⋊ L.

Structure of blocks with normal defect and abelian inertial quotient

“…The multiplication in (U Jen * (𝑃) 𝐻)𝑒 induces an isomorphism𝔄 ⊗ 𝑘 𝔐 − − → (U Jen * (𝑃) 𝐻)𝑒.Proof. The proof is similar to that of Theorem 4.15 of[4]. Applying Lemma 3.13 with 𝐴 = 𝔐 and B the subalgebra generated by 𝔄 and 𝔐, we see that the given map is injective.…”

“…As in [4], we now make use of the following lemma (see Chapter 3, Corollary 4.3 in Bass [1]). Lemma 3.13.…”

“…The purpose of this paper is to generalise the results from [4] to blocks of group algebras over k of finite groups that have a normal defect group P which is no longer necessarily abelian, but still with abelian 𝑝 inertial quotient L. By a theorem of Külshammer [13], any such block is isomorphic to a matrix algebra over a twisted group algebra 𝑘 𝛼 (𝑃 𝐿) of the semidirect product 𝑃 𝐿, for some 𝛼 ∈ 𝐻 2 (𝐿, 𝑘 × ), inflated to 𝑃 𝐿. So there is a central 𝑝 -extension Theorem 1.1.…”

Arbitrarily large Morita Frobenius numbers

“…Remark 6.2. From Proposition 5.8 and Proposition 6.1 (2) it follows that a τ as in Proposition 6.1 (2) will, up to an inner automorphism, satisfy either…”

“…In work of Benson, Kessar and Linckelmann [2, Theorem 1.1] the bound of two was extended to blocks that don't necessarily have a unique isomorphism class of simple modules. One can also define Morita Frobenius numbers over a complete discrete valuation ring O of characteristic zero with residue field k, and in [2] it was also shown that the aforementioned bound of two applies equally to the Morita Frobenius numbers of the corresponding blocks defined over O. Finally, Farrell [6, Theorem 1.1] and Farrell and Kessar [7,Theorem 1.1] proved that the Morita Frobenius number of any block of a finite quasi-simple group is at most four (both over k and over O).…”