Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles

Abstract

As part of a program to develop $K$-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas ${\overline{\mathfrak{A}}}_a={\sum }_b{Q}_b^a(\beta ){\overline{\mathfrak{P}}}_b$ and ${\overline{\mathfrak{Q}}}_a={\sum }_b{M}_b^a(\beta ){\overline{\mathfrak{F}}}_b,$ where each of ${\overline{\mathfrak{A}}}_a$, ${\overline{\mathfrak{P}}}_a$, ${\overline{\mathfrak{Q}}}_a$ and ${\overline{\mathfrak{F}}}_a$ is a family of polynomials that forms a basis for $\mathbb{Z}[x_1,\dots ,x_n][\beta ]$ indexed by weak compositions $a,$ and $Q_b^a(\beta )$ and $M_b^a(\beta )$ are monomials in $\beta$ for each pair $(a,b)$ of weak compositions. The polynomials ${\overline{\mathfrak{A}}}_a$ are the Lascoux atoms, ${\overline{\mathfrak{P}}}_a$ are the kaons, ${\overline{\mathfrak{Q}}}_a$ are the quasiLascoux polynomials, and ${\overline{\mathfrak{F}}}_a$ are the glide polynomials; these are respectively the $K$-analogues of the Demazure atoms ${\mathfrak{A}}_a$, the fundamental particles ${\mathfrak{P}}_a$, the quasikey polynomials ${\mathfrak{Q}}_a$, and the fundamental slide polynomials ${\mathfrak{F}}_a$. Monical, Pechenik, and Searles conjectured that for any fixed $a,$ ${\sum }_b{Q}_b^a(-1),{\sum }_b{M}_b^a(-1)\in \{0,1\},$ where $b$ ranges over all weak compositions. We prove this conjecture using a sign-reversing involution.

12 pages, comments welcome! v2: small edits to match journal version