Polynomials from combinatorial $K$-theory

TL;DR

This paper introduces two new polynomial bases related to combinatorial $K$-theory, establishing positive expansion relations and exploring their connections to known bases and cohomological analogues.

Contribution

It presents the quasiLascoux and kaon bases, providing positive expansion formulas and linking $K$-theoretic deformations to existing polynomial bases.

Findings

Positive expansions of quasiLascoux into glide and Lascoux atom bases

First proof of positive expansion of $K$-quasiSchur polynomials in multifundamental quasisymmetric polynomials

Positive expansions of glide and Lascoux atom bases into the kaon basis

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.