Top-degree components of Grothendieck and Lascoux polynomials
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135.

The CastelnuovoâMumford polynomial ${\stackrel{^}{\mathrm{đ}}}_{w}$ with $wâ{S}_{n}$ is the highest homogeneous component of the Grothendieck polynomial ${\mathrm{đ}}_{w}$. Pechenik, Speyer and Weigandt define a statistic $\mathsf{rajcode}\left(Â·\right)$ on ${S}_{n}$ that gives the leading monomial of ${\stackrel{^}{\mathrm{đ}}}_{w}$. We introduce a statistic $\mathsf{rajcode}\left(Â·\right)$ on any diagram $D$ through a combinatorial construction âsnow diagramâ that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation $w$, $\mathsf{rajcode}\left(D\right)$ agrees with the aforementioned $\mathsf{rajcode}\left(w\right)$. When $D$ is the key diagram of a weak composition $\mathrm{Î±}$, $\mathsf{rajcode}\left(D\right)$ yields the leading monomial of ${\stackrel{^}{\mathrm{đ}}}_{\mathrm{Î±}}$, the highest homogeneous component of the Lascoux polynomials ${\mathrm{đ}}_{\mathrm{Î±}}$. We use ${\stackrel{^}{\mathrm{đ}}}_{\mathrm{Î±}}$ to construct a basis of ${\stackrel{^}{V}}_{n}$, the span of ${\stackrel{^}{\mathrm{đ}}}_{w}$ with $wâ{S}_{n}$. Then we show ${\stackrel{^}{V}}_{n}$ gives a natural algebraic interpretation of a classical $q$-analogue of Bell numbers.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.326
Classification: 05E05
Keywords: Grothendieck polynomials, Lascoux polynomials, Hilbert series, CastelnuovoâMumford polynomials

Pan, Jianping 1; Yu, Tianyi 2

1 Department of Mathematics, NC State University, Raleigh, NC 95616-8633, U.S.A.
2 Department of Mathematics, UC San Diego, La Jolla, CA 92093, U.S.A.
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Pan, Jianping; Yu, Tianyi. Top-degree components of Grothendieck and Lascoux polynomials. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135. doi : 10.5802/alco.326. https://alco.centre-mersenne.org/articles/10.5802/alco.326/

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