Quasisymmetric functions and Kazhdan-Lusztig polynomials

Billera, Louis J.; Brenti, Francesco

doi:10.1007/s11856-011-0070-0

Cited by 33 publications

(50 citation statements)

References 31 publications

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“…However, since k > 1 this contradicts (2). Thus the edges of B s (u, v) are the cover relations of a poset.…”

Section: Proposition 3 By Ignoring the Directions Of The Edges B S (mentioning

confidence: 94%

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Shortest path poset of Bruhat intervals

Blanco

2012

J Algebr Comb

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We define the shortest path poset SP (u, v) of a Bruhat interval [u, v], by considering the shortest u-v paths in the Bruhat graph of a Coxeter group W , where u, v ∈ W . We consider the case of SP (u, v) having a unique rising chain under a reflection order and show that in this case SP(u, v) is a Gorenstein * poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.

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“…However, since k > 1 this contradicts (2). Thus the edges of B s (u, v) are the cover relations of a poset.…”

Section: Proposition 3 By Ignoring the Directions Of The Edges B S (mentioning

confidence: 94%

“…, P k , each one of which has a unique rising chain. A possible approach would be to "flip" the descents of a chain of SP(u, v) into ascents (see [2,Sect. 6]).…”

Section: Further Directionsmentioning

confidence: 99%

Shortest path poset of Bruhat intervals

Blanco

2012

J Algebr Comb

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“…We note here that if we were only interested in the complete cd-index of the interval [u, v], it could have been defined directly by means of a nonhomogeneous ab polynomial Ψ u,v defined analogously to (2), using the quantities b α in place of h S (see [12,Proposition 2.9]). However, the form of the quasisymmetric function Φ u,v given in Proposition 4.3 leads directly to a way of expressing the KazhdanLusztig polynomial P u,v in terms of the coefficients of the complete cd-index.…”

Section: Kazhdan-lusztig Polynomial and The Complete Cd-indexmentioning

confidence: 99%

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

Billera

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW -spheres, and Bruhat intervals in Coxeter groups. We review the two algebraic approaches to flag enumeration -one essentially as a quotient of the algebra of noncommutative symmetric functions and the other as a subalgebra of the algebra of quasisymmetric functions -and their relation via duality of Hopf algebras. One result is a direct expression for the Kazhdan-Lusztig polynomial of a Bruhat interval in terms of a new invariant, the complete cd-index. Finally, we summarize the theory of combinatorial Hopf algebras, which gives a unifying framework for the quasisymmetric generating functions developed here.Mathematics Subject Classification (2010). Primary 06A11; Secondary 05E05, 16T30, 20F55, 52B11.

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“…Billera and Brenti [2] associated the interval [u, v] with a non-commutative polynomialφ u,v (a, b) in the variables a and b. They further proved thatφ u,v (a, b) can be written as a polynomial in the variables c and d, where c = a + b, d = ab + ba.…”

Section: Introductionmentioning

confidence: 99%

On the non-negativity of the complete cd-index

Fan

2015

Discrete Mathematics

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a b s t r a c tThe complete cd-index of a Bruhat interval is a non-commutative polynomial in the variables c and d, which was introduced by Billera and Brenti and conjectured to have non-negative coefficients. For a cd-monomial M containing at most one d, i.e., M = c i or M = c i dc j (i, j ≥ 0), Karu showed that the coefficient of M is non-negative. In this paper, we show that when M = dc i dc j (i, j ≥ 0), the coefficient of M is non-negative.

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Quasisymmetric functions and Kazhdan-Lusztig polynomials

Cited by 33 publications

References 31 publications

Shortest path poset of Bruhat intervals

Shortest path poset of Bruhat intervals

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

On the non-negativity of the complete cd-index

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