$K$-analogues of Hivert's divided difference operators

TL;DR

This paper introduces $K$-analogues of Hivert's divided difference operators and demonstrates that the resulting polynomials include several important combinatorial and representation-theoretic families, extending previous $K$-theoretic frameworks.

Contribution

The authors define new $K$-analogues of Hivert's fundamental divided difference operators and identify the resulting polynomials with known combinatorial objects, unifying various families under a common $K$-theoretic framework.

Findings

Defined $K$-analogues of Hivert's fundamental divided difference operators.

Established that the polynomials from these operators are the multifundamentals, glides, and kaons.

Connected new operators to existing combinatorial and representation-theoretic polynomials.

Abstract

Several families of polynomials of combinatorial and representation theoretic interest (notably the Schur polynomials $s_\lambda$, Demazure characters $\mathfrak{D}_a$, and Demazure atoms $\mathfrak{A}_a$) can be defined in terms of divided difference operators. Hivert (2000) defines "fundamental analogues" of these divided difference operators, and Hivert and Hicks-Niese show in arXiv:2406.02420 that the polynomials that arise from those fundamental operators in analogous ways to the three families of polynomials above are respectively the fundamental quasisymmetric functions $F_a$ from (1984), the fundamental slides $\mathfrak{F}_a$ of Assaf and Searles from arXiv:1603.09744, and the fundamental particles $\mathfrak{P}_a$ of Searles from arXiv:1707.01172. Lascoux (2001) defines $K$-analogues of the divided difference operators, and in arXiv:1908.07364, Buciumas, Scrimshaw, and Weber show that the polynomials arising in corresponding ways from the $K$-theoretic divided difference operators are respectively the Grothendieck polynomials $\overline{s}_\lambda$, the combinatorial Lascoux polynomials $\overline{\mathfrak{D}}_a$ from arXiv:1611.08777, and the combinatorial Lascoux atoms $\overline{\mathfrak{A}}_a$ from arXiv:1611.08777, as conjectured by Monical in arXiv:1611.08777. We define $K$-analogues of Hivert's fundamental divided difference operators and show that the polynomials arising in the corresponding ways from our new operators are respectively the multifundamentals $\overline{F}_a$ of Lam and Pylyavskyy from arXiv:0705.2189, the fundamental glides $\overline{\mathfrak{F}}_a$ from of Pechenik and Searles from arXiv:1611.02545, and the kaons $\overline{\mathfrak{P}}_a$ of Monical, Pechenik, and Searles from arXiv:1806.03802.