A Major-Index Preserving Map on Fillings

Alexandersson, Per; Sawhney, Mehtaab

doi:10.37236/6893

Cited by 6 publications

(14 citation statements)

References 10 publications

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“…In particular, these expand positively into key polynomials with Kostka-Foulkes polynomials (in q) as coefficients. There are representation-theoretical explanations for these expansions, as well, see [2,3] and references therein for details.…”

Section: Discussionmentioning

confidence: 99%

“…Given a composition α, let par(α) be the unique integer partition where the parts of α have been rearranged in decreasing order. For example, par (2,0,1,4,9) is equal to (9, 4, 2, 1, 0). We can act with permutations on compositions (and partitions) by permutation of the entries:…”

Section: Bruhat Order Compositions and Operatorsmentioning

confidence: 99%

“…We use a similar approach to give bijections among families of coinversionfree fillings of general composition shapes in [2].…”

Section: Corollary 32mentioning

confidence: 99%

“…Our results are in some sense optimal: for general compositions λ, it happens that E σ λ (x; 1, t) cannot be factorized further. For example, Mathematica computations suggest that E (3,1,5,2,4) (0,2,3,1,0) (x; 1, t) and E (3,1,5,2,4) (0,1,1,1,0) (x; 1, 0) are irreducible.…”

Section: Lemma 43 We Have the Identitymentioning

confidence: 99%

“…The rows with labels 2 and 3 constitute a monotone pair and can be obtained using (5.2), which explains theπ 2 -arrow. Continuing on withπ 1 followed byθ 2 leads to an augmented diagram without any monotone pairs, so A 1342 3102 (x; t) = x (3,2,1,0) . Finally, following the arrows yields the operator expression:…”

Section: Positive Expansions At Q =mentioning

confidence: 99%

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Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

Alexandersson

Sawhney

2019

Ann. Comb.

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We examine the non-symmetric Macdonald polynomials E λ at q = 1, as well as the more general permuted-basement Macdonald polynomials. When q = 1, we show that E λ (x; 1, t) is symmetric and independent of t whenever λ is a partition. Furthermore, we show that, in general λ, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on x, t, and the relative order of the entries in λ. We also examine the case q = 0, which gives rise to the so-called permutedbasement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basementpermuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson.Mathematics Subject Classification. 05E10, 05E05.

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Section: Discussionmentioning

confidence: 99%

Section: Bruhat Order Compositions and Operatorsmentioning

confidence: 99%

“…We use a similar approach to give bijections among families of coinversionfree fillings of general composition shapes in [2].…”

Section: Corollary 32mentioning

confidence: 99%

Section: Lemma 43 We Have the Identitymentioning

confidence: 99%

Section: Positive Expansions At Q =mentioning

confidence: 99%

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Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

Alexandersson

Sawhney

2019

Ann. Comb.

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Demazure crystals and the Schur positivity of Catalan functions

Blasiak,

Morse,

Pun

2024

Invent. math.

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Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$ k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_{q}(\widehat{\mathfrak{sl}}_{\ell })$ U q ( sl ˆ ℓ ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals.

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Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials

Assaf¹

2018

Trans. Amer. Math. Soc.

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Abstract. We study the specialization of the type A nonsymmetric Macdonald polynomials at t " 0 based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of the Kostka-Foulkes polynomials.

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A Major-Index Preserving Map on Fillings

Cited by 6 publications

References 10 publications

Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

Properties of Non-symmetric Macdonald Polynomials at $$q=1$$ and $$q=0$$

Demazure crystals and the Schur positivity of Catalan functions

Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials

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