TY - JOUR
T1 - The Donald-Flanigan problem for finite reflection groups
AU - Gerstenhaber, Murray
AU - Giaquinto, Anthony
AU - Schaps, Mary E.
PY - 2001/4
Y1 - 2001/4
N2 - The Donald-Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra k H 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 wreath product Sn = H x ⋯ x H (n times) x Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H wreath product 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.
AB - The Donald-Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra k H 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 wreath product Sn = H x ⋯ x H (n times) x Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H wreath product 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.
KW - Coxeter groups
KW - Deformations
KW - Donald-Flanigan problem
KW - Finite reflection groups
KW - Finite representation type
KW - Weyl groups
KW - Wreath products
UR - http://www.scopus.com/inward/record.url?scp=0042404981&partnerID=8YFLogxK
U2 - 10.1023/A:1010846906745
DO - 10.1023/A:1010846906745
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AN - SCOPUS:0042404981
SN - 0377-9017
VL - 56
SP - 41
EP - 72
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
IS - 1
ER -