[Show abstract][Hide abstract] 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 G-invariant
polynomial functions on V to the algebra of C-invariant 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 G-orbits
in the G-saturation 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 Orlik-Solomon 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 Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional 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.
Transactions of the American Mathematical Society 10/2014; 366(10). DOI:10.1090/S0002-9947-2014-06060-1 · 1.12 Impact Factor
[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 Orlik-Solomon 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.
Journal of Algebra 03/2014; DOI:10.1016/j.jalgebra.2014.10.025 · 0.60 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We describe the equivariant K-groups 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 Kazhdan-Lusztig "bar"
involutions and Kazhdan-Lusztig bases for these equivariant K-groups.
Transformation Groups 02/2013; 19(1). DOI:10.1007/s00031-014-9257-x · 0.57 Impact Factor
[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 one-dimensional 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.
[Show abstract][Hide abstract] 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.
Journal of Algebra 01/2012; 377:320–332. DOI:10.1016/j.jalgebra.2012.11.047 · 0.60 Impact Factor
[Show abstract][Hide abstract] 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.
Communications in Algebra 01/2011; 41(7):2574-2592. DOI:10.1080/00927872.2012.661005 · 0.39 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We refine a conjecture by Lehrer and Solomon on the structure of the
Orlik-Solomon 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 Orlik-Solomon algebra of $W$ can be decomposed into a sum
of induced one-dimensional 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 $A$.
Transactions of the American Mathematical Society 01/2011; · 1.12 Impact Factor
[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 W-invariant polynomial functions on
V to the algebra of C-invariant 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.
Journal of Algebra 06/2009; 321(11-321):3158-3196. DOI:10.1016/j.jalgebra.2008.10.027 · 0.60 Impact Factor
[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 Borel-Moore 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 two-variable, 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 Kazhdan-Lusztig "R-polynomials" 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 R-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials for H that reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group when b=1. Comment: 25 pages; This is a substantive revision. A construction of a Kazhdan-Lusztig C basis and Kazhdan-Lusztig polynomials for H was added. A lot of typos were fixed. Some new typos might have been introduced
Representation Theory of the American Mathematical Society 10/2006; DOI:10.1090/S1088-4165-2010-00394-5 · 0.73 Impact Factor
[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 anti-invariants 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 $W_P \times W_Q$-anti-invariants of $H_{4n}(Z)$ are isomorphic to the top Borel-Moore 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 Borel-Moore 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 $W_P \times W_Q$-anti-invariants 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.
Transactions of the American Mathematical Society 06/2005; 360(11). DOI:10.1090/S0002-9947-08-04570-4 · 1.12 Impact Factor
[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.
Advances in Mathematics 10/2004; 187(2):396-416. DOI:10.1016/j.aim.2003.09.002 · 1.29 Impact Factor