Universal deformation rings for the symmetric group S_4

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.Comment: 12 pages, 2 figure

Minimal Resolutions of Algebras

AbstractA method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite-dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras, and Koszul algebras, with the aim of unifying existing treatments of these in the literature

Irreducible maps and bilinear forms

AbstractBautista showed in 1982 that the possible multiplicities of indecomposable summands of the domains and ranges of irreducible maps between modules over artin algebras are given by numerical invariants of certain bilinear forms associated with the algebra. We obtain further information about these multiplicities by relating the forms to those studied elsewhere in algebra and geometry. One spectacular result is that the allowable multiplicities for some algebras over the field of real numbers depend on J.F. Adams’ determination of the number of linearly independent vector fields on a sphere