35 research outputs found
Semiinvariants of Finite Reflection Groups
Let G be a finite group of complex n by n unitary matrices generated by
reflections acting on C^n. Let R be the ring of invariant polynomials, and \chi
be a multiplicative character of G. Let \Omega^\chi be the R-module of
\chi-invariant differential forms. We define a multiplication in \Omega^\chi
and show that under this multiplication \Omega^\chi has an exterior algebra
structure. We also show how to extend the results to vector fields, and exhibit
a relationship between \chi-invariant forms and logarithmic forms.Comment: Paper presented at 1999 Joint Meetings in San Antonio, special
session on Geometry in Dynamics. Typo correcte
Finite groups acting linearly: Hochschild cohomology and the cup product
When a finite group acts linearly on a complex vector space, the natural
semi-direct product of the group and the polynomial ring over the space forms a
skew group algebra. This algebra plays the role of the coordinate ring of the
resulting orbifold and serves as a substitute for the ring of invariant
polynomials from the viewpoint of geometry and physics. Its Hochschild
cohomology predicts various Hecke algebras and deformations of the orbifold. In
this article, we investigate the ring structure of the Hochschild cohomology of
the skew group algebra. We show that the cup product coincides with a natural
smash product, transferring the cohomology of a group action into a group
action on cohomology. We express the algebraic structure of Hochschild
cohomology in terms of a partial order on the group (modulo the kernel of the
action). This partial order arises after assigning to each group element the
codimension of its fixed point space. We describe the algebraic structure for
Coxeter groups, where this partial order is given by the reflection length
function; a similar combinatorial description holds for an infinite family of
complex reflection groups.Comment: 30 page
Poincare-Birkhoff-Witt Theorems
We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics.
Along the way, we compare modern techniques used to establish such results, for
example, the Composition-Diamond Lemma, Groebner basis theory, and the
homological approaches of Braverman and Gaitsgory and of Polishchuk and
Positselski. We discuss several contexts for PBW theorems and their
applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and
symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research
Institute Proceeding