共查询到20条相似文献,搜索用时 15 毫秒
1.
Pietro Mongelli 《Journal of Algebraic Combinatorics》2014,39(2):497-525
We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements. 相似文献
2.
R. M. Green 《Journal of Algebraic Combinatorics》2009,30(2):165-171
Let W be a Coxeter group of type
. We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P
x,w
is always equal to 0 or 1 if x is fully commutative (and w is arbitrary). 相似文献
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《Journal of Algebra》2003,259(2):613-627
In [J. Amer. Math. Soc. 5 (1992) 805–851] Stanley introduced the concept of a P-kernel for any locally finite partially ordered set P. In [Proc. Sympos. Pure Math., Vol. 56, AMS, 1994, pp. 135–148] Du introduced, for any set P, the concept of an IC basis. The purpose of this article is to show that, under some mild hypotheses, these two concepts are equivalent, and to characterize, for a given Coxeter group W, partially ordered by Bruhat order, the W-kernel corresponding to the Kazhdan–Lusztig basis of the Hecke algebra of W. Finally, we show that this W-kernel factorizes as a product of other W-kernels, and that these provide a solution to the Yang–Baxter equations for W. 相似文献
5.
《Indagationes Mathematicae》2021,32(6):1240-1274
We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers. 相似文献
6.
We provide simple rules for the computation of Kazhdan–Lusztig polynomials in the maximal parabolic case. They are obtained by filling regions delimited by paths with “Dyck strips” obeying certain rules. We compare our results with those of Lascoux and Schützenberger. 相似文献
7.
Michael Chmutov 《Journal of Algebraic Combinatorics》2015,42(4):1059-1076
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David Plaza 《代数通讯》2013,41(10):4354-4377
We obtain explicit branching rules for graded cell modules and graded simple modules over the endomorphism algebra of a Bott–Samelson bimodule. These rules allow us to categorify a well-known recursive formula for Kazhdan–Lusztig polynomials. 相似文献
10.
Brant C. Jones 《Journal of Algebraic Combinatorics》2009,29(2):229-260
We show that the leading coefficient of the Kazhdan–Lusztig polynomial P
x,w
(q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance
in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata
63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w.
The author received support from NSF grants DMS-9983797 and DMS-0636297. 相似文献
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We study the parabolic Kazhdan–Lusztig polynomials for the quasi-minuscule quotients of Weyl groups. We give explicit closed combinatorial formulas for the parabolic Kazhdan–Lusztig polynomials of type q. Our study implies that these are always either zero or a monic power of q, and that they are not combinatorial invariants. We conjecture a combinatorial interpretation for the parabolic Kazhdan–Lusztig polynomials of type −1. 相似文献
13.
A Newton–Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin–Fourier–Littelmann–Vinberg polytopes are examples of Newton–Okounkov convex bodies. In this paper, we prove that the Newton–Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima–Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara’s involution (\(*\)-operation) corresponds to a change of valuations on the rational function field. 相似文献
14.
William Y. C. Chen Neil J. Y. Fan Alan J. X. Guo Peter L. Guo Harry H. Y. Huang Michael X. X. Zhong 《Mathematische Zeitschrift》2014,277(3-4):1017-1025
In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\) -polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element. 相似文献
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《Discrete Mathematics》2007,307(19-20):2400-2414
17.
When \(\mathrm {Sp}(2n,\mathbb {C})\) acts on the flag variety of \(\mathrm {SL}(2n,\mathbb {C})\), the orbits are in bijection with fixed point free involutions in the symmetric group \(S_{2n}\). In this case, the associated Kazhdan–Lusztig–Vogan polynomials \(P_{v,u}\) can be indexed by pairs of fixed point free involutions \(v\ge u\), where \(\ge \) denotes the Bruhat order on \(S_{2n}\). We prove that these polynomials are combinatorial invariants in the sense that if \(f:[u,w_0]\rightarrow [u',w_0]\) is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then \(P_{v,u} = P_{f(v),u'}\) for all \(v\ge u\). 相似文献
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We formulate a strong positivity conjecture on characters afforded by the Alvis–Curtis dual of the intersection cohomology of Deligne–Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent ?-blocks of finite reductive groups. 相似文献
20.
Neil J.Y. Fan Peter L. Guo Grace L.D. Zhang 《Journal of Pure and Applied Algebra》2017,221(1):237-250
Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let be the symmetric group on , and let be the generating set of , where for , is the adjacent transposition. For a subset , let be the parabolic subgroup generated by J, and let be the set of minimal coset representatives for . For in the Bruhat order and , let denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for when , and obtained an expression for when . In this paper, we provide a formula for , where and i appears after in v. It should be noted that the condition that i appears after in v is equivalent to that v is a permutation in . We also pose a conjecture for , where with and v is a permutation in . 相似文献