Publications (15)11.25 Total impact
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ABSTRACT: Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is the fixed point subspace of an element of G. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of Ginvariant polynomial functions on V to the algebra of Cinvariant functions on X. Extending earlier work by Douglass and Roehrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups G in terms of the exponents of G and C, and their reflection arrangements. A consequence of our main result is that the variety of Gorbits in the Gsaturation of X is smooth if and only if it is normal.  [Show abstract] [Hide abstract]
ABSTRACT: We refine a conjecture by Lehrer and Solomon on the structure of the OrlikSolomon algebra of a finite Coxeter group W and relate it to the descent algebra of W. As a result, we claim that both the group algebra of W and the OrlikSolomon algebra of W can be decomposed into a sum of induced onedimensional representations of element centralizers, one for each conjugacy class of elements of W. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair (W,W L ), where W is arbitrary and W L is a parabolic subgroup of W, all of whose irreducible factors are of type A.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the p th graded component of its OrlikSolomon algebra as a sum of characters induced from linear characters of centralizers of elements of W for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.  [Show abstract] [Hide abstract]
ABSTRACT: We describe the equivariant Kgroups of a family of generalized Steinberg varieties that interpolates between the Steinberg variety of a reductive, complex algebraic group and its nilpotent cone in terms of the extended affine Hecke algebra and double cosets in the extended affine Weyl group. As an application, we use this description to define KazhdanLusztig "bar" involutions and KazhdanLusztig bases for these equivariant Kgroups.  [Show abstract] [Hide abstract]
ABSTRACT: In our recent paper (Douglass et al. arXiv:1101.2075 (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik–Solomon algebra of W can be decomposed into a sum of induced onedimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2. 
Article: Computations for Coxeter arrangements and Solomon's descent algebra II: Groups of rank five and six
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ABSTRACT: In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the graded components of its Orlik–Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W. The refined conjecture relates the character above to a decomposition of the regular character of W related to Solomonʼs descent algebra of W. The refined conjecture has been proved for symmetric and dihedral groups, as well as for finite Coxeter groups of rank three and four. In this paper, we prove the conjecture for finite Coxeter groups of rank five and six. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik–Solomon characters of the groups considered. 
Article: Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four
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ABSTRACT: In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group W afforded by the homogeneous components of its Orlik–Solomon algebra as sums of characters induced from linear characters of centralizers of elements of W. Our refined conjecture also relates the Orlik–Solomon characters above to the terms of a decomposition of the regular character of W related to the descent algebra of W. A consequence of our conjecture is that both the regular character of W and the character of the Orlik–Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of W, one for each conjugacy class of elements of W. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik–Solomon characters of these groups.  [Show abstract] [Hide abstract]
ABSTRACT: Let $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter elements to be injective, up to conjugacy.  [Show abstract] [Hide abstract]
ABSTRACT: We refine a conjecture by Lehrer and Solomon on the structure of the OrlikSolomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the OrlikSolomon algebra of $W$ can be decomposed into a sum of induced onedimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_L$ is a parabolic subgroup of $W$ all of whose irreducible factors are of type $A$.  [Show abstract] [Hide abstract]
ABSTRACT: Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of Winvariant polynomial functions on V to the algebra of Cinvariant functions on X. In this note we consider the special case when W is a Coxeter group, V is the complexified reflection representation of W, and X is in the lattice of the arrangement of W, and give a simple, combinatorial characterization of when the restriction mapping is surjective in terms of the exponents of W and C. As an application of our result, in the case when W is the Weyl group of a semisimple, complex, Lie algebra, we complete a calculation begun by Richardson in 1987 and obtain a simple combinatorial characterization of regular decomposition classes whose closure is a normal variety.  [Show abstract] [Hide abstract]
ABSTRACT: We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck. The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne–Langlands conjecture for Hecke algebras by Kazhdan and Lusztig.  [Show abstract] [Hide abstract]
ABSTRACT: Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded BorelMoore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W. Comment: 17 pages, to appear in Documenta Math  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we define a twovariable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q) induced from a solvable subgroup. We construct KazhdanLusztig "Rpolynomials" for H and show that they may be used to define a partial order on G(b,1,n). Using a generalization of Deodhar's notion of distinguished subexpressions we give a closed formula for the Rpolynomials. After passing to a onevariable quotient of the ring of scalars, we construct KazhdanLusztig polynomials for H that reduce to the usual KazhdanLusztig polynomials for the symmetric group when b=1. Comment: 25 pages; This is a substantive revision. A construction of a KazhdanLusztig C basis and KazhdanLusztig polynomials for H was added. A lot of typos were fixed. Some new typos might have been introduced  [Show abstract] [Hide abstract]
ABSTRACT: Let $G$ be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and antiinvariants of the cohomology of the Springer fibres of the cone of nilpotent elements, $\mathcal N$, of $\operatorname{Lie}(G)$ for the Steinberg variety $Z$ of triples. Using a general specialization argument we show that for a parabolic subgroup $W_P \times W_Q$ of $W \times W$ the space of $W_P \times W_Q$invariants and the space of $W_P \times W_Q$antiinvariants of $H_{4n}(Z)$ are isomorphic to the top BorelMoore homology groups of certain generalized Steinberg varieties introduced in [5]. The rational group algebra of the Weyl group $W$ of $G$ is isomorphic to the opposite of the top BorelMoore homology $H_{4n}(Z)$ of $Z$, where $2n = \dim \mathcal N$. Suppose $W_P \times W_Q$ is a parabolic subgroup of $W \times W$. We show that the space of $W_P \times W_Q$invariants of $H_{4n}(Z)$ is $e_Q{\mathbb Q} We_P$, where $e_P$ is the idempotent in group algebra of $W_P$ affording the trivial representation of $W_P$ and $e_Q$ is defined similarly. We also show that the space of $W_P \times W_Q$antiinvariants of $H_{4n}(Z)$ is $\epsilon_Q{\mathbb Q} W\epsilon_P$, where $\epsilon_P$ is the idempotent in group algebra of $W_P$ affording the sign representation of $W_P$ and $\epsilon_Q$ is defined similarly.  [Show abstract] [Hide abstract]
ABSTRACT: For a reductive, algebraic group, G, the Steinberg variety of G is the set of all triples consisting of a unipotent element, u, in G and two Borel subgroups of G that contain u. We define generalized Steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the Weyl group. Using one of the two special cases, we define a parabolic version of a map from the Weyl group to a set of nilpotent orbits of G in Lie(G) defined by Joseph and study some of its properties.
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43  Citations  
11.25  Total Impact Points  
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20042014

University of North Texas
 Department of Mathematics
Denton, Texas, United States
