1,720,982 research outputs found
On a duality in Coxeter groups
No full textIn [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 Poincaré series, and by a formal argument in terms of inclusion-exclusion-like matrices. © 2007 Elsevier Ltd. All rights reserved
Closed Product Formulas for CertainR-polynomials
No full textAbstractR -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
The combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials of lower intervals
Full text linkBoolean elements in Kazhdan–Lusztig theory
No full textAbstractKazhdan–Lusztig polynomials have been proven to play an important role in different fields. Despite this, there are still few explicit formulae for them. Here we give closed product formulae for the Kazhdan–Lusztig polynomials indexed by Boolean elements in a class of Coxeter systems that we call linear. Boolean elements are elements smaller than a reflection that admits a reduced expression of the form s1…sn−1snsn−1…s1 (si∈S, si≠sj if i≠j). Then we provide several applications of this result concerning the combinatorial invariance of the Kazhdan–Lusztig polynomials, the classification of the pairs (u,v) with u≺v, the Kazhdan–Lusztig elements and the intersection homology Poincaré polynomials of the Schubert varieties
Root Polytopes and Borel Subalgebras
Full text linkLet Φ be a finite crystallographic irreducible root system and PΦ be the convex hull of the roots in Φ. We give a uniform explicit description of the polytope PΦ, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results
Special matchings in Coxeter groups
No full textSpecial 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
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