Grothendieck to Lascoux expansions

Shimozono, Mark; Yu, Tianyi

doi:10.48550/arxiv.2106.13922

Cited by 3 publications

(5 citation statements)

References 24 publications

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“…, }. In this context our formula is an equivariant upgrade of the symmetric Grothendieck special case of [SY,Theorem 1.3]. This result is also implicit in [BSW].…”

Section: K-bumpless Pipedreamsmentioning

confidence: 73%

Back stable K-theory Schubert calculus

Lam¹,

Lee²,

Shimozono³

2021

Preprint

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We study the back stable K-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double K-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to K-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials, and divided difference operators in K-homology.

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“…, }. In this context our formula is an equivariant upgrade of the symmetric Grothendieck special case of [SY,Theorem 1.3]. This result is also implicit in [BSW].…”

Section: K-bumpless Pipedreamsmentioning

confidence: 73%

Back stable K-theory Schubert calculus

Lam¹,

Lee²,

Shimozono³

2021

Preprint

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“…Finally, Reiner and Yong [RY21] conjectured an expansion of Grothendieck polynomials into Lascoux polynomials, generalizing expansions in both [RS95] and [BKS + 08]. Shimozono and Yu [SY21] proved this conjecture.…”

Section: Introductionmentioning

confidence: 88%

“…1 Notice that we replace "smallest" by "largest" and "at least" by "at most". This is because [SY21] focused on reverse SSYT (tableaux whose rows are weakly decreasing and columns are strictly decreasing) while this paper focused on SSYT.…”

Section: Compute Right Keys Using the Star Operatormentioning

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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“…Compute right keys using the star operator. Shimozono and Yu [SY21] used the following operator to compute right keys.…”

Section: The Right Keysmentioning

confidence: 99%

Set-valued tableaux rule for Lascoux polynomials

Yu¹

2021

Preprint

Self Cite

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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 rule for Lascoux polynomials involving right keys and set-valued tableaux. Our rule recovers the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our rule, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw in the context of abstract Kashiwara crystal.

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Grothendieck to Lascoux expansions

Cited by 3 publications

References 24 publications

Back stable K-theory Schubert calculus

Back stable K-theory Schubert calculus

Set-valued tableaux rule for Lascoux polynomials

Set-valued tableaux rule for Lascoux polynomials

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