TL;DR
This paper introduces genomic tableaux as a new combinatorial framework that extends classical tableau theory to derive novel Littlewood-Richardson rules for K-theory Schubert calculus in various Grassmannian types.
Contribution
It develops genomic tableaux as a semistandard complement to increasing tableaux, leading to new combinatorial rules and conjectures in Schubert calculus.
Findings
Derived new Littlewood-Richardson rules for Grassmannians and orthogonal Grassmannians.
Established genomic tableaux as a tool for K-theoretic Schubert calculus.
Proposed conjectures for Lagrangian Grassmannians based on bounds from genomic tableaux.
Abstract
We explain how genomic tableaux [Pechenik-Yong '15] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood-Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch '02]) and maximal orthogonal Grassmannians (after [Clifford-Thomas-Yong '14], [Buch-Ravikumar '12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.
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