• Gerstehenhaber Brackets for Skew Group Algebras, with Sarah Witherspoon.
  Submitted.  Abstract:  
  Hochschild cohomology  governs deformations of algebras, and its graded Lie structure plays a vital role. We study this structure for the Hochschild cohomology of the skew group algebra formed by a finite group acting on an algebra by automorphisms.We examine the Gerstenhaber bracket with a view toward deformations and developing  bracket formulas. We then focus on the linear group actions and polynomial algebras that arise in orbifold theory and representation theory; deformations in this context include graded Hecke algebras and symplectic reflection algebras. We  give some general results describing when brackets are zero for polynomial skew group algebras, which allow us in particular to find noncommutative Poisson structures. For abelian groups, we express the bracket using inner products of group characters. Lastly, we interpret results for graded Hecke algebras. 

• Finite groups actings linearly: Hochschild cohomology and the cup product, with Sarah Witherspoon.
  Submitted.  Abstract:  
  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.

• Quantum differentiation and chain maps of bimodule complexes, with Sarah Witherspoon.  
  Submitted.   Abstract:
  We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space.  This skew group algebra illuminates the resulting orbifold and serves as a replacement for the ring of invariant polynomials, especially in the eyes of cohomology.  One analyzes the Hochschild cohomology of the skew group algebra using isomorphisms which convert between resolutions. We present an explicit chain map from the bar resolution to the Koszul resolution of the symmetric algebra which induces various isomorphisms on Hochschild homology and cohomology, some of which have appeared in the literature before. This approach unifies previous results on homology and cohomology of both the symmetric algebra and skew group algebra. We determine induced combinatorial cochain maps which invoke quantum differentiation (expressed by Demazure-BBG operators).

•  Jacobians of Reflection Groups. With Julia Hartmann.  
  Transactions of the American Mathematical Society, 360 (2008), no.1, 123--133.  Abstract: 
  Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes.  This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields, using a ramification formula of Benson and Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.

• Hochschild cohomology and graded Hecke algebras, with Sarah Witherspoon. 
  Transactions of the American Mathematical Society 360 (2008), no. 8, 3975--4005. Abstract:
  We develop and collect techniques for determining Hochschild cohomology of skew group algebras S(V) # G and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups G(r,p,n) to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of S(V) # G and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of G on V, and we show that there exist nontrivial graded Hecke algebras for G(r,1,n), in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

•  Reflection groups and differential forms, with Julia Hartmann.  
  Mathematical Research Letters 14 (2007), no.6, 955--971.  Abstract:
  Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for groups with polynomial ring of invariants over arbitrary fields using a ramification formula of Benson and Crawley-Boevey.

• Generalized Exponents and Forms.  dvi  
  Journal of Algebraic Combinatorics, 22 (2005), no. 1, 115--132.  Abstract:
  We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer's theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.

•  Graded Hecke Algebras for Complex Reflection Groups, with Arun Ram. 
  Commentairi M. Helvetici, 78, 308-334, 2003. dvi,  ps.  Abstract:
  The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence.  A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V).  This paper classifies all the algebras obtained by applying Drinfeld's construction to complex reflection groups.  By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld's construction.  The classification shows that there exist algebras obtained from Drinfeld's construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups.

•  The Sign representation for Shephard Groups, with Peter Orlik and Victor Reiner. 
  Mathematische Annalen, 322, p. 477-492, 2002. ps, dvi  Abstract:
  Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes.  For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes.  This representation turns out to have many equivalent guises making it analogous to the sign representation of a finite Coxeter group.  One of these guises is (up to a twist) the cohomology of the Milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invariant of minimal degree. 
  

• Logarithmic forms and anti-invariant forms of reflection groups, with Hiroaki Terao.    In Advanced Studies in Pure Math., 27, Arrangements, Tokyo, 1998, 2000,  pdf, ps  Abstract:
  Let W be a finite group generated by unitary reflections and A be the set of reflecting hyperplanes.  We will give a characterization of the logarithmic differential forms with poles along A in terms of anti-invariant differential forms.  If W is a Coxeter group defined over the real numbers, then the characterization provides a new method to find a basis for the module of logarithmic differential forms out of basic invariants.

• Semi-invariants of finite reflection groups.  pdf, dvi. 
  Journal of Algebra 220, 314-326 (1999).  Online at IDEAL.  Abstract:
  
Let G be a finite group of complex n x n unitary matrices generated by reflections acting on C^n.  Let R be the ring of invariant polynomials, and let \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.

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