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 ask s for a deformation of the group algebra kH to a separable algebra. It is s olved 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 gr oups 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 an d if S-n is the symmetric group then (i) the problem is solvable also for t he wreath product H S-n = H x ... x H (n times) x S-n and (ii) given a morp hism from a finite Abelian or dihedral group G to S-n it is solvable also f or H G. The theorems suggested by the Donald-Flanigan conjecture and subseq uently proven follow, we also show, from a geometric conjecture which altho ugh weaker for groups applies to a broader class of algebras than group alg ebras.