Two-sided cells in type BB (asymptotic case)

We compute two-sided cells of Weyl groups of type $B$ for the "asymptotic" choice of parameters. We also obtain some partial results concerning Kazhdan-Lusztig conjectures in this particular case.Comment: 20 pages, some misprints have been cleaned up in this second versio

The generalized lifting property of Bruhat intervlas

In [E. Tsukerman and L. Williams, {\em Bruhat Interval Polytopes}, Advances in Mathematics, 285 (2015), 766-810] it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.Comment: 18 page

Special matchings in Coxeter groups

Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan--Lusztig polynomials.Comment: 19 page

Pattern avoidance and the Bruhat order on involutions

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.Comment: 14 pages, 5 figure

Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters

Hecke algebras arise naturally in the representation theory of reductive groups over finite or p-adic fields.  These algebras are specializations of Iwahori-Hecke algebras which can be defined in terms of a Coxeter group and a weight function without reference to reductive groups and this is the setting we are working in.  Kazhdan-Lusztig cells play a crucial role in the study of Iwahori-Hecke algebras.  The aim of this work is to study the Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters.  More precisely, we show that the Kazhdan-Lusztig polynomials of an affine Weyl group are invariant under “long enough” translations, we decompose the lowest two-sided cell into left cells and we determine the decomposition of the affine Weyl group of type Ğ2 into cells for a whole class of weight functions.EThOS - Electronic Theses Online ServiceGBUnited Kingdo