Top-degree components of Grothendieck and Lascoux polynomials

TL;DR

This paper introduces a combinatorial statistic called jcode, applicable to diagrams and permutations, to identify leading monomials of top-degree components of Grothendieck and Lascoux polynomials, linking algebraic and combinatorial structures.

Contribution

It defines a new jcode statistic on diagrams that generalizes previous permutation-based statistics, providing a unified combinatorial approach to leading monomials of polynomial components.

Findings

jcode correctly identifies leading monomials for Grothendieck and Lascoux polynomials.

Constructs a basis of the span of top-degree Grothendieck polynomials.

Provides an algebraic interpretation of a q-analogue of Bell numbers.

Abstract

The Castelnuovo-Mumford polynomial $\widehat{\mathfrak{G}}_w$ with $w \in S_n$ is the highest homogeneous component of the Grothendieck polynomial $\mathfrak{G}_w$. Pechenik, Speyer and Weigandt define a statistic $\mathsf{rajcode}(\cdot)$ on $S_n$ that gives the leading monomial of $\widehat{\mathfrak{G}}_w$. We introduce a statistic $\mathsf{rajcode}(\cdot)$ on any diagram $D$ through a combinatorial construction ``snow diagram'' that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation $w$, $\mathsf{rajcode}(D)$ agrees with the aforementioned $\mathsf{rajcode}(w)$. When $D$ is the key diagram of a weak composition $\alpha$, $\mathsf{rajcode}(D)$ yields the leading monomial of $\widehat{\mathfrak{L}}_\alpha$, the highest homogeneous component of the Lascoux polynomials $\mathfrak{L}_\alpha$. We use $\widehat{\mathfrak{L}}_\alpha$ to construct a basis of $\widehat{V}_n$, the span of $\widehat{\mathfrak{G}}_w$ with $w \in S_n$. Then we show $\widehat{V}_n$ gives a natural algebraic interpretation of a classical $q$-analogue of Bell numbers.