The Donald–Flanigan Problem for Finite Reflection Groups

Abstract

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types Bn and Dn (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if Sn is the symmetric group then (i) the problem is solvable also for the wreath product H\(\wr\)Sn = H × ··· × H (n times) ⋊ Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H\(\wr\)G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.