Blocks inequivalent to their Frobenius twists

Abstract

Let k be an algebraically closed field of characteristic p. We give a general method for producing examples of blocks B of finite group algebras that are not Morita equivalent as k-algebras to the Frobenius twist B (p) . Our method produces non-nilpotent blocks having one simple module and elementary abelian defect group. These also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive, but no derived equivalence between the blocks.We do not know of any examples of blocks B that are not Morita equivalent to the second Frobenius twist B (p 2 ) .

BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

“…The purpose of this paper is to bound the Frobenius numbers and to give a structure theorem for blocks of finite groups with normal abelian defect groups and abelian p ′ inertial quotients. This extends the results of Benson and Green [2], Holloway and Kessar [7], Benson and Kessar [3].…”

On stable equivalences and blocks with one simple module

“…In the context of a block algebra B which has, up to isomorphism, a unique simple module S, it may be tempting to think that S would play that role. However, even the smallest known and well understood example in [3,4] of a non-nilpotent block with one isomorphism class of simple modules has the property that its block cohomology is not isomorphic to the Ext-algebra of a module over that block, and this is what we are going to describe in this section. Let k be an algebraically field of odd prime characteristic p and let P = C p × C p be an elementary abelian group of order p 2 .…”

Rationality of blocks of quasi-simple finite groups

“…In many cases in the above theorem we show that the Morita Frobenius number of B is in fact equal to 1 (see Theorem 8.1), and in no case do we show that the number is greater than 1. We note that there are examples of blocks of ℓ-solvable groups with Morita Frobenius number equal to 2 [2]. In [18], the first author calculated the Morita Frobenius numbers of k ⊗ O B for several families of blocks B of quasi-simple finite groups.…”