1,720,974 research outputs found
Right-angled Coxeter groups, universal graphs, and Eulerian polynomials
Full text linkWe define a class of representations for any right-angled Coxeter group R and its Hecke algebra H(R). The group R and the algebra H(R) act on any Bruhat interval of any Coxeter system (W,S), once given a suitable function from the set of Coxeter generators of R to the power set of S. The existence of such a function is related to the problem of universality of a graph G2 constructed from the unlabeled Coxeter graph G of (W,S). When G=Pn is a path with n vertices, we conjecture that G2 is n-universal; this property is equivalent to the existence of an action of R and H(R) on the Bruhat intervals of the symmetric group Sn+1, for all right-angled groups with n generators. We prove that (Pn)2 is n-universal for forests. Eulerian polynomials arise as characters of our representations, when the Coxeter graph of R is a path. We also give a formula for the toric h-polynomial of any lower Bruhat interval in a universal Coxeter group Un, using results on the Kazhdan–Lusztig basis of H(Un)
Special idempotents and projections
Full text linkWe define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call special idempotent any idempotent element of this monoid. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings
Parabolic Temperley-Lieb modules and polynomials
No full textWe define and study, for any Coxeter system (W, S), modules over its Temperley-Lieb algebra, two for each quotient WJ, which have generators indexed by the fully commutative elements of WJ. Our results are new even in type A and include, for J=θ, those obtained in [16] and [17]
Artin group injection in the Hecke algebra for right-angled groups
Full text linkWe prove some injectivity results: that a Coxeter monoid Z-algebra (or 0-Hecke algebra) injects in the incidence Z-algebra of the corresponding Bruhat poset, for any Coxeter group; that the Hecke algebra of a right-angled Coxeter group injects in the Coxeter monoid Z[q, q- 1] -algebra (and then in the incidence Z[q, q- 1] -algebra of the corresponding Bruhat poset); that a right-angled Artin group injects in the group of invertible elements of the Hecke algebra of the corresponding Coxeter group (and then in the group of invertible elements of a Coxeter monoid algebra and in the one of an incidence algebra)
Parabolically induced functions and equidistributed pairs
Full text linkGiven a function defined over a parabolic subgroup of a Coxeter group, equidistributed with the length, we give a procedure to construct a function over the entire group, equidistributed with the length. Such a procedure permits to define functions equidistributed with the length in all the finite Coxeter groups. We can establish our results in the general setting of graded posets which satisfy some properties. These results apply to some known functions arising in Coxeter groups as the major index, the negative major index and the D-negative major index defined in type A, B, and D, respectively
Complements of Coxeter group quotients
No full textWe consider the complement WWJ of any quotient WJ of a Coxeter system (W,S) and we investigate its algebraic, combinatorial and geometric properties, emphasizing its connection with parabolic Kazhdan–Lusztig theory. In particular, we define two families of polynomials which are the analogues, for the poset WWJ, of the parabolic Kazhdan–Lusztig and R-polynomials. These polynomials, indexed by elements of WWJ, have interesting connections with the ordinary Kazhdan–Lusztig and R-polynomials
Isomorphisms of hecke modules and parabolic kazhdan-lusztig polynomials
No full textWe define and study some Hecke modules which generalize the ones defined by Deodhar in [10] and we find isomorphisms between them. As consequences, we obtain equalities between parabolic Kazhdan-Lusztig polynomials of different Coxeter groups. In particular, we show that any parabolic Kazhdan-Lusztig polynomial equals a parabolic polynomial of a maximal quotient, and we find equalities between the parabolic polynomials of quasi-minuscule quotients and those of certain maximal quotients of the corresponding affine Weyl groups. © 2014 Elsevier Inc
Odd and even major indices and one-dimensional characters for classical Weyl groups
Full text linkWe define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs
The Jones-Wenzl idempotent of a generalized Temperley-Lieb algebra
No full textWe define an idempotent of the Temperley-Lieb algebra of any finite Coxeter system, which reduces, in type A, to the known Jones-Wenzl idempotent. We give, for any pair of parabolic subgroups, a recursive formula generalizing the well-known one. This approach gives a wide class of recursive formulas for the classical Jones-Wenzl idempotent. We also compute explicitly the coefficient corresponding to the maximal element of any minuscule quotient, when the idempotent is expressed in the basis of fully commutative elements
Parabolic Kazhdan-Lusztig R-polynomials for quasi-minuscule quotients
Full text linkWe give explicit combinatorial formulas for the parabolic
Kazhdan-Lusztig R-polynomials of the quasi-minuscule quotients of the
classical Weyl groups. As an
application of our results we obtain explicit combinatorial
formulas for certain sums and alternating sums of ordinary
Kazhdan-Lusztig R-polynomials
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