Asymmetric Function Theory

Pechenik, Oliver; Searles, Dominic

doi:10.1007/978-981-15-7451-1_5

Cited by 4 publications

(4 citation statements)

References 39 publications

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“…bijection [22] semistandard key tableaux ← −−−−−−− → the same [2] semistandard skyline fillings Figure 1.1: Relations among some combinatorial models of key polynomials.…”

Section: Kohnert Tableauxmentioning

confidence: 99%

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An Involution on Semistandard Skyline Fillings

Fan,

Guo,

Liu

2020

Preprint

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Non-attacking skyline fillings were used by Haglund, Haiman and Loehr to establish a combinatorial formula for nonsymmetric Macdonald polynomials. Semistandard skyline fillings are non-attacking skyline fillings with both major index and coinversion number equal to zero, which serve as a combinatorial model for key polynomials. In this paper, we construct an involution on semistandard skyline fillings. This involution can be viewed as a vast generalization of the classical Bender-Knuth involution. As an application, we obtain that semistandard skyline fillings are compatible with the Demazure operators, offering a new combinatorial proof that nonsymmetric Macdonald polynomials specialize to key polynomials.

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“…bijection [22] semistandard key tableaux ← −−−−−−− → the same [2] semistandard skyline fillings Figure 1.1: Relations among some combinatorial models of key polynomials.…”

Section: Kohnert Tableauxmentioning

confidence: 99%

“…Assaf [2] introduced semistandard key tableaux to give another combinatorial construction of key polynomials. Comparing the definitions of Kohnert tableaux [4] and semistandard key tableaux [2], it can be checked that Kohnert tableaux are in bijection with semistandard key tableaux [22].…”

Section: Introductionmentioning

confidence: 99%

An Involution on Semistandard Skyline Fillings

Fan,

Guo,

Liu

2020

Preprint

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“…As the name suggests, quasisymmetric Schur functions share many properties with classical Schur functions, and Mason's map ρ defined in [24] connects the combinatorics of composition tableaux to that of Young tableaux. Understanding analogues of Schur functions and their generalizations has long been a theme in algebraic combinatorics; see [1,2,3,4,5,25,28,33] for recent work in this context.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative LR coefficients and crystal reflection operators

Richmond¹,

Tewari²

2019

Preprint

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“…See [2,5,6,11] for some remarkable aspects of these symmetric functions and related topics. Given the recent interest in polynomial analogues of combinatorially-defined quasisymmetric and symmetric functions [13], it is natural to investigate chromatic quasisymmetric functions. For the class of graphs mentioned earlier, a nonsymmetric polynomial analogue was proposed by Haglund-Wilson [10], and this is the chief object of our study.…”

Section: Introductionmentioning

confidence: 99%

Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive

Tewari¹,

Wilson²,

Zhang³

2019

Preprint

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Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the chromatic nonsymmetric polynomial of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged (P, ρ)-partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel's fundamental basis by taking a backstable limit of our expansion.

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Asymmetric Function Theory

Cited by 4 publications

References 39 publications

An Involution on Semistandard Skyline Fillings

An Involution on Semistandard Skyline Fillings

Noncommutative LR coefficients and crystal reflection operators

Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive

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