Type A molecules are Kazhdan–Lusztig

Chmutov, Michael

doi:10.1007/s10801-015-0616-z

Cited by 7 publications

(14 citation statements)

References 6 publications

(5 reference statements)

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“…While as shown by Blasiak [6] getting exactly the right axiomatization to address those questions can be very challenging, Assaf's work provides a very useful framework. In particular her characterization of dual equivalence graphs has been used in a variety of contexts, see for example Chmutov [9] and Roberts [18]. Assaf's ideas were further developed by Blasiak-Fomin [5] and others.…”

Section: Introductionmentioning

confidence: 99%

Robinson–Schensted correspondence for unit interval orders

Kim

Pylyavskyy

2021

Sel. Math. New Ser.

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The Stanley-Stembridge conjecture associates a symmetric function to each natural unit interval order P. In this paper, we define relations à la Knuth on the symmetric group for each P and conjecture that the associated P-Knuth equivalence classes are Schur-positive, refining theorems of Gasharov, Brosnan-Chow, Guay-Paquet, and Shareshian-Wachs. The resulting equivalence graphs fit into the framework of D graphs studied by Assaf. Furthermore, we conjecture that the Schur expansion is given by column-readings of P-tableaux that occur in the equivalence class. We prove these conjectures for P avoiding two specific suborders by introducing P-analog of Robinson-Schensted insertion, giving an answer to a long standing question of Chow.

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Section: Introductionmentioning

confidence: 99%

Robinson–Schensted correspondence for unit interval orders

Kim

Pylyavskyy

2021

Sel. Math. New Ser.

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“…It follows from Remark 6.23 that if µ is the empty partition then the dual equivalence graph is isomorphic to the simple part of each Kazhdan-Lusztig left cell Γ(C(t)) for t ∈ Std(λ ); in this case we call the dual equivalence graph the standard dual equivalence graph corresponding to λ ∈ P(n). Extending earlier work of Assaf [1], Chmutov showed in [4] that the simple part of an admissible W n -molecule is isomorphic to a standard dual equivalence graph. The following result is the main theorem of [4].…”

Section: Tableaux Left Cells and Admissible Molecules Of Type Amentioning

confidence: 65%

“…Extending earlier work of Assaf [1], Chmutov showed in [4] that the simple part of an admissible W n -molecule is isomorphic to a standard dual equivalence graph. The following result is the main theorem of [4]. THEOREM 6.40.…”

Section: Tableaux Left Cells and Admissible Molecules Of Type Amentioning

confidence: 65%

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Type $A$ admissible cells are Kazhdan–Lusztig

Nguyễn¹

2020

Algebraic Combinatorics

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Admissible W-graphs were defined and combinatorially characterised by Stembridge in [12]. The theory of admissible W-graphs was motivated by the need to construct W-graphs for Kazhdan-Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan-Lusztig polynomials. In this paper, we shall show that type A-admissible W-cells are Kazhdan-Lusztig as conjectured by Stembridge in his original paper.the elements of which are called the left descents of w and the right descents of w, respectively. Kazhdan and Lusztig give a recursive procedure that defines polynomials P y,w whenever y, w ∈ W and y < w. These polynomials satisfy deg P y,w 1 2 (l(w) − l(y) − 1), and µ y,w is defined to be the leading coefficient of P y,w if the degree is 1 2 (l(w) − l(y) − 1), or 0 otherwise.to be the group opposite to W, and observe that (W ×W o , S S o ) is a Coxeter system. Kazhdan and Lusztig show that if µ and τ are defined by the formulasgraph. Thus M = AW may be regarded as an (H, H)-bimodule. Furthermore, the construction produces an explicit (H, H)-bimodule isomorphism M ∼ = H. It follows easily from the definition of µ y,w that µ(y, w) = 0 only if l(w) − l(y) is odd; thus (W, µ,τ) is a bipartite graph. The non-negativity of all coefficients of the Kazhdan-Lusztig polynomials, conjectured in [8], has been proved by Elias and Williamson in [5]. Since W and W o are standard parabolic subgroups of W ×W o , it follows that Γ = (W, µ, τ) is a W-graph and Γ o = (W, µ, τ o ) is a W o -graph, where τ and τ o are defined by τ(w) = L(w) and τ o (w) = R(w) o , for all w ∈ W .In accordance with the theory described above, there are preorders on W determined by the (W ×W o )-graph structure, the W-graph structure and the W o -graph structure. We call these the two-sided preorder (denoted by LR ), the left preorder ( L ) and the right preorder ( R ). The corresponding cells are the two-sided cells, the left cells and the right cells. ADMISSIBLE W-GRAPHSLet (W, S) be a Coxeter system, not necessarily finite. For s, t ∈ S, let m(s,t) be the order of st in W. Thus {s,t} is a bond in the Coxeter diagram if and only if m(s,t) > 2.

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“…Variations of dual equivalence graphs have also been given for k-Schur functions in [Assaf and Billey, 2012], for the product of a Schubert polynomial with a Schur polynomial in , and for shifted tableaux in relation to the type B Lie group in [Billey et al, 2014] and [Assaf, 2014]. Dual equivalence graphs were also connected to Kazhdan-Lusztig polynomials and W -graphs by Michael Chmutov in [Chmutov, 2013]. This paper will focus on dual equivalence graphs that emerge as components of a larger family of graphs.…”

Section: Introductionmentioning

confidence: 99%

On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

Roberts

2017

Electron. J. Combin.

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This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H δ (X; q, t) and H δ (X; 0, t), respectively. We then give an explicit Schur expansion of H δ (X; 0, t) as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function R γ,δ (X) as a refinement of H δ (X; 0, t) and similarly describe its Schur expansion. We then analyze R γ,δ (X) to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of H δ (X; q, t). To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph H δ . In the case where a subgraph of H δ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries. * Partially supported by DMS-1101017 from the NSF.1. P δ := (V ′ , σ, E ′ ) is the subgraph with V ′ = {π ∈ S n : inv δ (π) = 0}.

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Type A molecules are Kazhdan–Lusztig

Cited by 7 publications

References 6 publications

Robinson–Schensted correspondence for unit interval orders

Robinson–Schensted correspondence for unit interval orders

Type $A$ admissible cells are Kazhdan–Lusztig

On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

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