Abstract. Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincaré-Birkhoff-Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.

Abstract. Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. While these partial orders are of independent combinatorial interest, our investigation is motivated by a connection between the codimension order and the algebraic structure of cohomology governing deformations of skew group algebras. In this article, we compare the reflection length and codimension functions and discuss implications for cohomology of skew group algebras. We give algorithms using character theory for computing reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and computer calculations in GAP, we show that Coxeter groups and the infinite family G(m, 1, n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the infinite family G(m, p, n), which immediately yields an explicit description of generators for cohomology.

ABSTRACT. Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.

No abstract

No abstract