Top-degree components of Grothendieck and Lascoux polynomials

Abstract

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.