The Donald-Flanigan problem for finite reflection groups -: To the memory of Moshe!Flato z"l

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 B-n and D-n (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 S-n is the symmetric group then (i) the problem is solvable also for the wreath product H S-n = H x ... x H (n times) x S-n and (ii) given a morphism from a finite Abelian or dihedral group G to S-n it is solvable also for H 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.