We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M , in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We al...

Proofs that endomorphism (convolution) algebras are commutative most often depend upon identifying an anti-involution to interchange the order of factors, but which nevertheless acts as the identity on the algebra or suitable subalgebras. Silberger gave such an argument for the spherical Hecke algebras of p-adic reductive groups, for example. Apparently the first occurrence of the Gelfand-Kazhd...

Four statistics, ls, rb, rs, and lb, previously studied on all partitions of { 1, 2, ..., n }, are applied to non-crossing partitions. We consider single and joint distributions of these statistics and prove equidistribution results. We obtain qand p, q-analogues of Catalan and Narayana numbers which refine the rank symmetry and unimodality of the lattice of non-crossing partitions. Two unimoda...

In this paper, we investigate the structure and representations of the quantum group U(∞) = Uυ(gl∞). We will present a realization for U(∞), following Beilinson–Lusztig–MacPherson (BLM) [1], and show that the natural algebra homomorphism ζr from U(∞) to the infinite q-Schur algebra S(∞, r) is not surjective for any r > 1. We will give a BLM type realization for the image U(∞, r) := ζr(U(∞)) and...

Let W be a Coxeter group of type ̃ An−1. We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial Px,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced concept of matroids. It is known that both matroid can be considered as special cases Kazhdan-Lusztig-Stanley locally finite posets. framework polynomials, we study inverse functions define We also compute these boolean matroids uniform As an unexpected application obtain a ne...

Let H be the Iwahori–Hecke algebra associated with Sn, the symmetric group on n symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. While the former admits a deep geometric interpretation, the latter leads to a purely combinatorial construction of the representations of H, including the Dipper–James theory of Specht modules. In this paper, we establish...

In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the c...

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 KazhdanLusztig polynomial Pid,w(q) = 1 + q for some h. Résumé. On démontre la conjecture de Billey et Braden sur les permutations w pour lesquelles le polynôme de Kazhdan-Lusztig Pid,w(q) = 1...