A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

Alexandersson, Per; Sulzgruber, Robin

doi:10.48550/arxiv.2004.09198

Cited by 4 publications

(5 citation statements)

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“…We now relate increasing spanning forests of an indifference graph G to orientations of G, following [Ale20] and [AS20]. We say that an oriented edge − → uv of G is oriented to the right if u < v, and oriented to the left otherwise.…”

Section: Relation With the Chromatic Quasisymmetric Function And Llt ...mentioning

confidence: 99%

A symmetric function of increasing forests

Abreu,

Nigro

2020

Preprint

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For an indifference graph G we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular LLT polynomials. As a consequence we give a combinatorial interpretation of the coefficients of the LLT polynomial in the elementary basis (up to a factor of a power of (q − 1)), strengthening the description given in [AS20].

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Section: Relation With the Chromatic Quasisymmetric Function And Llt ...mentioning

confidence: 99%

A symmetric function of increasing forests

Abreu,

Nigro

2020

Preprint

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“…Building from this intuition, it is possible to define the operation of plethysm in a way that extends to a definition f [g] valid for any symmetric functions f and g. This operation is fundamental to modern symmetric function theory research; many problems and results are best stated and studied through plethystic simplifications, such as questions related to LLT polynomials [1,2], Macdonald polynomials [8,11], and Hall-Littlewood polynomials [12,13].…”

Section: Introductionmentioning

confidence: 99%

Plethysms of Chromatic and Tutte Symmetric Functions

Crew¹,

Spirkl²

2021

Preprint

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Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question.In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.

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“…D'Adderio [5] used recurrences in terms of Schröder paths to prove that the shifted verticalstrip LLT polynomial G λ (x; q + 1) is a positive linear combination of elementary symmetric functions. Alexandersson conjectured [1] and then proved with Sulzgruber [2] the explicit combinatorial formula (1.4) G λ (x; q + 1) = θ∈O(P )…”

Section: Introductionmentioning

confidence: 99%

A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

Tom¹

2020

Preprint

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In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial G λ (x; q) in some special cases. We associate a weighted graph Π to λ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of G λ (x; q) whenever Π is triangle-free. We also prove that the largest power of q in the LLT polynomial is the total edge weight of our graph.

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A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

Cited by 4 publications

References 0 publications

A symmetric function of increasing forests

A symmetric function of increasing forests

Plethysms of Chromatic and Tutte Symmetric Functions

A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials

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