Mario Marietti

Closed Product Formulas for CertainR-polynomials

R -polynomials get their importance from the fact that they are used to define and compute the Kazhdan?Lusztig polynomials, which have applications in several fields. Here we give a closed product formula for certainR -polynomials valid for every Coxeter group. This result implies a conjecture due to F. Brenti about the symmetric groups.

Diamonds and Hecke algebra representations

A class of partially ordered sets called diamonds, that includes all Coxeter groups ordered by Bruhat order, is introduced. It is shown that the definition of Kazhdan- Lusztig polynomials can be generalized to the framework of diamonds and that they can be used to construct a family of Hecke algebra representations that includes those constructed by Kazhdan and Lusztig and contains several new ones.

Formulas for multi-parameter classes of Kazhdan-Lusztig polynomials in S(n)

We give explicit formulas for several classes of Kazhdan-Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of Brenti and Simion [Explicit formulae for some Kazhdan-Lusztig polynomials, J. Algebraic Combin. 11 (2000) 187-196], we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of F. Caselli [Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials, J. Algebraic Combin. 18 (2003) 171-187].

Conical and spherical graphs

We introduce and study the notions of conical and spherical graphs. We show that these mutually exclusive properties, which have a geometric interpretation, provide links between apparently unrelated classical concepts such as dominating sets, independent dominating sets, edge covers, and the homotopy type of an associated simplicial complex. In particular, we solve the problem of characterizing the forests whose dominating sets of minimum cardinality are also independent. To establish these connections, we prove a formula to compute the Euler characteristic of an arbitrary simplicial complex from a set of generators of its Stanley-Reisner ideal.

On a duality in Coxeter groups

In [R.P. Stanley, The descent set and connectivity set of a permutation, J. Integer Seq. 8 (3) (2005) Article 05.3.8] Stanley gives certain enumerative identities revealing a duality between descent sets and connectivity sets of the symmetric group. In this paper we generalize these identities to all Coxeter groups. The proofs are obtained by giving these identities an algebraic explanation in terms of parabolic subgroups, coset representatives, and Poincare series, and by a formal argument in terms of inclusion-exclusion-like matrices.