Top-degree components of Grothendieck and Lascoux polynomials

Pan, Jianping; Yu, Tianyi

doi:10.5802/alco.326

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.

Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135.

Abstract

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

Received: 2023-03-08
Revised: 2023-07-11
Accepted: 2023-07-18
Published online: 2024-02-22

DOI: 10.5802/alco.326

Classification: 05E05
Keywords: Grothendieck polynomials, Lascoux polynomials, Hilbert series, Castelnuovo–Mumford polynomials

Author's affiliations:

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.

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Top-degree components of {Grothendieck} and {Lascoux} polynomials},
     journal = {Algebraic Combinatorics},
     pages = {109--135},
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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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