Let R be a ring generated by l elements with stable range r. Assume that the group ELd(R) has Kazhdan constant ǫ0 > 0 for some d ≥ r + 1. We prove that there exist ǫ(ǫ0, l) > 0 and k ∈ N, s.t. for every n ≥ d, ELn(R) has a generating set of order k and a Kazhdan constant larger than ǫ. As a consequence, we obtain for SLn(Z) where n ≥ 3, a Kazhdan constant which is independent of n w.r.t generat...

We provide a combinatorial proof for the coincidence of Knuth equivalence classes, Kazhdan–Lusztig left cells and Vogan classes for the symmetric group, involving only Robinson-Schensted algorithm and the combinatorial part of the Kazhdan–Lusztig cell theory. The determination of Kazhdan–Lusztig cells for the symmetric group is given in the proof of [4, Thm1.4]. The argument is largely combinat...

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...

In this article we improve the known Kazhdan constant for SLn(Z) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by [42 √ n + 860], which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since √ 2/n is an upper bound. We can use this bound to improve the bounds for the spectral gap of the ...

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1, . . . , xn,n] to construct irreducible Sn-modules. This construction produces exactly the same matrices as the Kazhdan-Lusztig construction [Invent.Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. It also produces exactly the same modules as those which Clausen constructed using a different basis ...

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusz...

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...

Let R be a ring generated by l elements with stable range r. Assume that the group EL d (R) has Kazhdan constant ǫ 0 > 0 for some d ≥ r + 1. We prove that there exist ǫ(ǫ 0 , l) > 0 and k ∈ N, s.t. for every n ≥ d, EL n (R) has a generating set of order k and a Kazhdan constant larger than ǫ. As a consequence, we obtain for SL n (Z) where n ≥ 3, a Kazhdan constant which is independent of n w.r....

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank n+1 thagomizer matroid by showing that the coefficient of tk is equal to the number of Dyck paths of semilength n with k long ascents. We also give a conjecture for the Sn-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.