We introduce a new multiplication in the incidence algebra of a partially ordered set, and study the resulting algebra. As an application of the properties of this algebra we obtain a combinatorial formula for the Kazhdan-Lusztig-Stanley functions of a poset. As special cases this yields new combinatorial formulas for the parabolic and inverse parabolic Kazhdan-Lusztig polyno-mials, for the gen...

We give explicit combinatorial product formulas for the maximal parabolic Kazhdan-Lusztig and R-polynomials of the symmetric group. These formulas imply that these polynomials are combinatorial invariants, and that the KazhdanLusztig ones are nonnegative. The combinatorial formulas are most naturally stated in terms of Young’s lattice, and the one for the Kazhdan-Lusztig polynomials depends on ...

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the pos-itivity of certain structure constants for the associated Kazhdan–Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading ter...

We give upper and lower bounds for the Kazhdan-Lusztig polynomials of any Coxeter group W. If W is nite we prove that, for any k 0, the k-th coeecient of the Kazhdan-Lusztig polynomial of two elements u, v of W is bounded from above and below by a polynomial (which depends only on k) in l(v)?l(u). In particular, this implies the validity of Lascoux-Schutzenberger's conjecture for all suuciently...

We find an explicit formula for the Kazhdan-Lusztig polynomials Pui,a ,vi of the symmetric group S(n) where, for a, i, n ∈ N such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = sasa+1 · · · si−1 and by vi the element ofS(n) obtained by inserting n in position i in any permutation ofS(n −1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two c...

We give simple examples of finitely presented Kazhdan groups with infinite outer automorphism groups, as arithmetic lattices in Lie groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answer...

We give simple examples of finitely presented Kazhdan groups with infinite outer automorphism groups, as arithmetic lattices in Lie groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answer...

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the pos-itivity of certain structure constants for the associated Kazhdan–Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading ter...

In their fundamental paper [18] 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 (see, e.g., [17], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the geometry of Schubert varieties, and have proven to be of fundamental importanc...