Alternating subgroups of Coxeter groups

We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.     

Odd length for even hyperoctahedral groups and signed generating functions

We define a new statistic on the even hyperoctahedral groups which is a natural analogue of the odd length statistic recently defined and studied on Coxeter groups of types $A$ and $B$. We compute the signed (by length) generating function of this statistic over the whole group and over its maximal and some other quotients and show that it always factors nicely. We also present some conjectures.     

Special matchings and Coxeter groups

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.] All authors are partially supported by EU grant HPRN-CT-2001-00272. Received: 30 October 2006     

Special matchings and Kazhdan-Lusztig polynomials

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, 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 polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials.     

Combinatorial Expansions of Kazhdan-Lusztig Polynomials

We introduce two related families of polynomials, easily computableby simple recursions into which any Kazhdan–Lusztig (andinverse Kazhdan–Lusztig) polynomial of any Coxeter groupcan be expanded linearly, and we give combinatorial interpretationsto the coefficients in these expansions. This yields a combinatorialrule for computing the Kazhdan–Lusztig polynomials interms of paths in a directed graph, and a completely combinatorialreformulation of the nonnegativity conjecture [15, p. 166].     

Explicit Formulae for Some Kazhdan-Lusztig Polynomials

We consider the Kazhdan-Lusztig polynomials P _u,v (q) indexed by permutations u, v having particular forms with regard to their monotonicity patterns. The main results are the following. First we obtain a simplified recurrence relation satisfied by P _u,v (q) when the maximum value of v S_n occurs in position n – 2 or n – 1. As a corollary we obtain the explicit expression for P_{e,3 4 ... n 1 2}(q) (where e denotes the identity permutation), as a q-analogue of the Fibonacci number. This establishes a conjecture due to M. Haiman. Second, we obtain an explicit expression for P_{e, 3 4 ... (n – 2) n (n – 1) 1 2}(q). Our proofs rely on the recurrence relation satisfied by the Kazhdan-Lusztig polynomials when the indexing permutations are of the form under consideration, and on the fact that these classes of permutations lend themselves to the use of induction. We present several conjectures regarding the expression for P _u,v (q) under hypotheses similar to those of the main results.     

The intersection cohomology of Schubert varieties is a combinatorial invariant

We give an explicit and entirely poset-theoretic way to compute, for any permutation v, all the Kazhdan–Lusztig polynomials P_x,y for x,y≤v, starting from the Bruhat interval [e,v] as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells.     

Quasisymmetric functions and Kazhdan-Lusztig polynomials

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.