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arXiv · 2110.00164

Set-valued tableaux rule for Lascoux polynomials

Tianyi Yu
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Abstract

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 Sch\"{u}tzenberger 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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Tianyi Yu. 2021-10-01. Set-valued tableaux rule for Lascoux polynomials. https://arxiv.org/abs/2110.00164

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