Polynomials from Combinatorial $K$-theory

Monical, Cara ; Pechenik, Oliver ; Searles, Dominic

Canadian journal of mathematics, Tome 73 (2021) no. 1, pp. 29-62

Résumé

We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$-theoretic deformation of the quasi-key basis and also a lift of the $K$-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric 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.

Détail

DOI : 10.4153/S0008414X19000464

Mots-clés : Demazure character, Demazure atom, Lascoux polynomial, Lascoux atom, Grothendieck polynomial, quasi-Lascoux polynomial, kaon