31 research outputs found
The Cohomology of the Regular Semisimple Variety
AbstractWe use the equivariant cohomology of hyperplane complements and their toral counterparts to give formulae for the Poincaré polynomials of the varieties of regular semisimple elements of a reductive complex Lie group or Lie algebra. As a result, we obtain vanishing theorems for certain of the Betti numbers. Similar methods, usingl-adic cohomology, may be used to compute numbers of rational points of the varieties over the finite field Fq. In the classical cases, one obtains, both for the Poincaré polynomials and for the numbers of rational points, polynomials which exhibit certain regularity conditions as the dimension increases. This regularity may be interpreted in terms of functional equations satisfied by certain power series, or in terms of the representation theory of the Weyl group
Generalised Euler Characteristics of Varieties of Tori in Lie Groups
Generalised Euler Characteristics of Varieties of Tori in Lie Groups
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
When W is a finite reflection group, the noncrossing partition lattice NCP_W
of type W is a rich combinatorial object, extending the notion of noncrossing
partitions of an n-gon. A formula (for which the only known proofs are
case-by-case) expresses the number of multichains of a given length in NCP_W as
a generalised Fuss-Catalan number, depending on the invariant degrees of W. We
describe how to understand some specifications of this formula in a case-free
way, using an interpretation of the chains of NCP_W as fibers of a
Lyashko-Looijenga covering (LL), constructed from the geometry of the
discriminant hypersurface of W. We study algebraically the map LL, describing
the factorisations of its discriminant and its Jacobian. As byproducts, we
generalise a formula stated by K. Saito for real reflection groups, and we
deduce new enumeration formulas for certain factorisations of a Coxeter element
of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation.
Version 3 : corrected typos, added illustrated example. To appear in Journal
of Algebraic Combinatoric
On centralizer algebras for spin representations
We give a presentation of the centralizer algebras for tensor products of
spinor representations of quantum groups via generators and relations. In the
even-dimensional case, this can be described in terms of non-standard
q-deformations of orthogonal Lie algebras; in the odd-dimensional case only a
certain subalgebra will appear. In the classical case q = 1 the relations boil
down to Lie algebra relations
A quantum analogue of the first fundamental theorem of invariant theory
We establish a noncommutative analogue of the first fundamental theorem of
classical invariant theory. For each quantum group associated with a classical
Lie algebra, we construct a noncommutative associative algebra whose underlying
vector space forms a module for the quantum group and whose algebraic structure
is preserved by the quantum group action. The subspace of invariants is shown
to form a subalgebra, which is finitely generated. We determine generators of
this subalgebra of invariants and determine their commutation relations. In
each case considered, the noncommutative modules we construct are flat
deformations of their classical commutative analogues. Thus by taking the limit
as , our results imply the first fundamental theorem of classical
invariant theory, and therefore generalise them to the noncommutative case.Comment: 44 pages, 3 figure