Top-degree components of Grothendieck and Lascoux polynomials

Jianping Pan, Tianyi Yu

Published 2023-02-07Version 1

We define a filtered algebra $\widehat{V_1} \subset \widehat{V_2} \subset \cdots \subset \widehat{V} \subset \mathbb{Q}[x_1, x_2, \dots]$ which gives an algebraic interpretation of a classical $q$-analogue of Bell numbers. The space $\widehat{V_n}$ is the span of the Castelnuovo-Mumford polynomials $\widehat{\mathfrak{G}}_w$ with $w \in S_n$. Pechenik, Speyer and Weigandt define $\widehat{\mathfrak{G}}_w$ as the top-degree components of the Grothendieck polynomials and extract a basis of $\widehat{V_n}$. We describe another basis consisting of $\widehat{\mathfrak{L}}_\alpha$, the top-degree components of Lascoux polynomials. Our basis connects the Hilbert series of $\widehat{V_n}$ and $\widehat{V}$ to rook-theoretic results of Garsia and Remmel. To understand $\widehat{\mathfrak{L}}_\alpha$, we introduce a combinatorial construction called a ``snow diagram'' that augments and decorates any diagram $D$. When $D$ is the key diagram of $\alpha$, its snow diagram yields the leading monomial of $\widehat{\mathfrak{L}}_\alpha$. When $D$ is the Rothe diagram of $w$, its snow diagram yields the leading monomial of $\widehat{\mathfrak{G}}_w$, agreeing with the work of Pechenik, Speyer and Weigandt.