Let (W, S) be a finite Coxeter system and A := Z[0] be the group algebra of a finitely generated free abelian group 0. Let H be an Iwahori-Hecke algebra of (W, S) over A with parameters vs . Further let K be an extension field of the field of fractions of A and KH be the extension of scalars. In this situation Kazhdan and Lusztig have defined their famous basis and the so called left cell modul...

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial Pid,w(q) = 1 + q for some h.

We classify the \fully tight" simply-laced Coxeter groups, that is, the ones whose iji-avoiding Kazhdan{Lusztig basis elements are monomials in the generators B s i. We then investigate the basis of the Temperley{Lieb algebra arising from the Kazhdan{Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements ofW , which have become known as the Kazhdan-Lusztig polynomials of W , and which have proven to be of importance in several areas of mathematics. In this paper we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these pol...

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhda...

We study two bases of the vector space of immanants of C[x1,1, . . . , xn,n]: the bitableaux basis of Désarménien-Kung-Rota, and a subset of the dual canonical basis called the basis of Kazhdan-Lusztig immanants. We show that the transition matrix between these bases is unitriangular, describe new vanishing results for the Kazhdan-Lusztig immanants, and relate both bases to other immanants defi...

Macdonald defined two-parameter Kostka functions Kλμ(q, t) where λ, μ are partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture that also these more general Kostka functions are polynomials in q and t with non-negative integers as coefficients. If q = 0 then our Kostka functions are Kazhdan-Lusztig polyn...

Let Uq be the quantum group associated to a Lie algebra g of rank n. The negative part U q of Uq has a canonical basis B with favourable properties (see Kashiwara [3] and Lusztig [6, §14.4.6]). The approaches of Lusztig and Kashiwara lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We describe the ...

We consider a class of relatively hyperbolic groups in the sense of Gromov and use an argument modeled after Carlsson–Pedersen to prove Novikov conjectures for these groups. This proof is related to [16, 17] which dealt with arithmetic lattices in rank one symmetric spaces and some other arithmetic groups of higher rank. Here we view the rank one lattices in this different larger context of rel...

This paper shows how the Kazhdan-Lusztig theory of cells can be directly applied to establish the quasi-heredity of q-Schur algebras. The application arises because of a very strong homological property enjoyed by certain cell filtrations for q-permutation modules.