Set-Valued Skyline Fillings

Abstract

Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart's Schur expansion of the Grothendieck polynomial $G_{\lambda }$, using the uncrowding operator of V. Reiner, B. Tenner, and A. Yong.

17 pages

References in corpus (4)

• Quasisymmetric Schur functions
• A Decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth Algorithm
• Genomic Tableaux
• Poset edge densities, nearly reduced words, and barely set-valued tableaux

Cited by in corpus (14)

• Genomic Tableaux
• A Demazure crystal construction for Schubert polynomials
• Crystal structures for canonical Grothendieck functions
• K-theoretic crystals for set-valued tableaux of rectangular shapes
• Newton Polytopes in Algebraic Combinatorics
• A crystal on decreasing factorizations in the $0$-Hecke monoid
• Asymmetric function theory
• The "Grothendieck to Lascoux" conjecture
• The genomic Schur function is fundamental-positive
• Grothendieck to Lascoux expansions
• Newton polytopes and symmetric Grothendieck polynomials
• Set-valued tableaux rule for Lascoux polynomials
• Hybrid Pipe Dreams for Key Polynomials
• An Involution on Semistandard Skyline Fillings