An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

Adve, Anshul; Robichaux, Colleen; Yong, Alexander

doi:10.1016/j.aim.2021.107669

Cited by 6 publications

(14 citation statements)

References 9 publications

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“…In this section, we study the support of top Lascoux polynomials. Our main tool is an elegant description of the support of Schubert polynomials given by Adve, Robichaux, and Yong [ARY21]. Definition 6.1.…”

Section: Support Of Top Lascoux Polynomialsmentioning

confidence: 99%

“…Definition 6.1. [ARY21] Let D be a diagram and α be a weak composition. Then PerfectTab Ó pD, αq is the set of fillings of D satisfying all of the following:…”

Section: Support Of Top Lascoux Polynomialsmentioning

confidence: 99%

“…(2) Lam, Lee and Shimozono [LLS21] introduced the (reduced) bumpless pipedreams (BPD) to compute the monomial expansion of Schubert polynomials. (3) Adve, Robichaux, and Yong [ARY21] introduced perfect tableaux to compute the support of Schubert polynomials. (4) The Schubert polynomials have the saturated Newton polytope (SNP) property [FMSD18].…”

Section: Introductionmentioning

confidence: 99%

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Set-valued tableaux rule for Lascoux polynomials

Yu¹

2023

Combinatorial Theory

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Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.

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Section: Support Of Top Lascoux Polynomialsmentioning

confidence: 99%

“…Definition 6.1. [ARY21] Let D be a diagram and α be a weak composition. Then PerfectTab Ó pD, αq is the set of fillings of D satisfying all of the following:…”

Section: Support Of Top Lascoux Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Set-valued tableaux rule for Lascoux polynomials

Yu¹

2023

Combinatorial Theory

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“…#Red(w 0 ) = f (n−1,n−2,..., 3,2,1) ; (4) hence #Red(w 0 ) is computed by (3). One measure of the brevity of ( 2) is the Edelman-Greene statistic on S n , EG(w) = !…”

Section: Reduced Word Combinatoricsmentioning

confidence: 99%

Reduced Word Enumeration, Complexity, and Randomization

Monical¹,

Pankow²,

Yong

2022

Electron. J. Combin.

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A reduced word of a permutation w is a minimal length expression of w as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is typically exponentially large. This implies a result of B. Pawlowski, that it has exponentially growing expectation. Our result is established by a formal run-time analysis of A. Lascoux and M. P. Schützenberger's transition algorithm. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.

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“…This completes the proof of both theorems. For the Schubert coefficients, see [Knu16,KZ17,Man01,MPP14] for positive combinatorial interpretations in several special cases, and [ARY21] for complexity of a related problem. For the row character sums a λ defined in (1.2.1), Frumkin [Fru86] proved that a λ ≥ 1 for all |λ| > 1.…”

mentioning

confidence: 99%

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Ikenmeyer,

Pak,

Panova

2023

International Mathematics Research Notices

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We prove that deciding the vanishing of the character of the symmetric group is $\textsf{C}_= \textsf{P}$-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in $\textsf{#P}$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof, we conclude that deciding positivity of the character is $\textsf{PP}$-complete under many-one reductions, and hence $\textsf{PH}$-hard under Turing reductions.

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An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

Cited by 6 publications

References 9 publications

Set-valued tableaux rule for Lascoux polynomials

Set-valued tableaux rule for Lascoux polynomials

Reduced Word Enumeration, Complexity, and Randomization

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

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