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.

30 pages, color figures

References in corpus (3)

• Set-Valued Skyline Fillings
• Schubert calculus and shifting of interval positroid varieties
• Poset edge densities, nearly reduced words, and barely set-valued tableaux

Cited by in corpus (19)

• Set-Valued Skyline Fillings
• Crystal structures for canonical Grothendieck functions
• Decompositions of Grothendieck Polynomials
• Crystal structures for symmetric Grothendieck polynomials
• Polynomials from combinatorial $K$-theory
• Orbits of Plane Partitions of Exceptional Lie Type
• Asymmetric function theory
• Shifted insertion algorithms for primed words
• The genomic Schur function is fundamental-positive
• Poset modules of the $0$-Hecke algebras and related quasisymmetric power sum expansions
• Combinatorial formulas for shifted dual stable Grothendieck polynomials
• Expanding K-theoretic Schur Q-functions
• Unique rectification in $d$-complete posets: towards the $K$-theory of Kac-Moody flag varieties
• Puzzles in $K$-homology of Grassmannians
• Shifted combinatorial Hopf algebras from $K$-theory
• Vanishing of Littlewood-Richardson polynomials is in P
• Shifted Bender-Knuth moves and a shifted Berenstein-Kirillov group
• Crystals for set-valued decomposition tableaux
• A shifted Berenstein-Kirillov group and the cactus group