Top-degree components of Grothendieck and Lascoux polynomials

Abstract

The Castelnuovo-Mumford polynomial ${\hat{\mathfrak{G}}}_w$ 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 ${\hat{\mathfrak{G}}}_w$. 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 $\hat{I}\pm$, $\mathsf{rajcode}(D)$ yields the leading monomial of ${\hat{\mathfrak{L}}}_{\hat{I}}\pm$, the highest homogeneous component of the Lascoux polynomials ${\mathfrak{L}}_{\hat{I}}\pm$. We use ${\hat{\mathfrak{L}}}_{\hat{I}}\pm$ to construct a basis of ${\hat{V}}_n$, the span of ${\hat{\mathfrak{G}}}_w$ with $w\in S_n$. Then we show ${\hat{V}}_n$ gives a natural algebraic interpretation of a classical $q$-analogue of Bell numbers.

26 pages; revision after getting referee report