Polynomials from combinatorial $K$-theory

Abstract

We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasiLascoux basis, which is simultaneously both a $K$-theoretic deformation of the quasikey basis and also a lift of the $K$-analogue of the quasiSchur basis from quasisymmetric polynomials to general polynomials. We give positive expansions of this quasiLascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasiLascoux basis. As a special case, these expansions give the first proof that the $K$-analogues of quasiSchur polynomials expand positively in multifundamental quasisymmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$-analogues mirror the relationships among their cohomological counterparts. We make several 'alternating sum' conjectures that are suggestive of Euler characteristic calculations.

arXiv comment35 pages, 10 figures

References in corpus (7)

• Quasisymmetric Schur functions
• Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
• Kohnert tableaux and a lifting of quasi-Schur functions
• Genomic Tableaux
• A Thom-Porteous formula for connective K-theory using algebraic cobordism
• Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials
• Decompositions of Grothendieck Polynomials

Cited by in corpus (6)

• Colored five-vertex models and Lascoux polynomials and atoms
• Crystal structures for canonical Grothendieck functions
• K-theoretic crystals for set-valued tableaux of rectangular shapes
• Crystal structures for symmetric Grothendieck polynomials
• Hybrid Pipe Dreams for Key Polynomials
• Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles