We give a new formula for double Grothendieck polynomials based on Magyar’s orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak{S}_w(\mathbf{x};\mathbf{y})$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n \mathfrak{S}_w(x_n^{-1}, \dots , x_1^{-1}; 1,\dots ,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.
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Keywords: Grothendieck polynomials, Lascoux polynomials
Setiabrata, Linus 1 ; St. Dizier, Avery 2
CC-BY 4.0
Setiabrata, Linus; St. Dizier, Avery. Double orthodontia formulas and Lascoux positivity. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 953-972. doi: 10.5802/alco.499
@article{ALCO_2026__9_4_953_0,
author = {Setiabrata, Linus and St. Dizier, Avery},
title = {Double orthodontia formulas and {Lascoux} positivity},
journal = {Algebraic Combinatorics},
pages = {953--972},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {4},
doi = {10.5802/alco.499},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.499/}
}
TY - JOUR AU - Setiabrata, Linus AU - St. Dizier, Avery TI - Double orthodontia formulas and Lascoux positivity JO - Algebraic Combinatorics PY - 2026 SP - 953 EP - 972 VL - 9 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.499/ DO - 10.5802/alco.499 LA - en ID - ALCO_2026__9_4_953_0 ER -
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