Frozen pipes: lattice models for Grothendieck polynomials

Brubaker, Ben; Frechette, Claire; Hardt, Andrew; Tibor, Emily; Weber, Katherine

doi:10.5802/alco.277

Brubaker, Ben ¹; Frechette, Claire ²; Hardt, Andrew ³; Tibor, Emily ¹; Weber, Katherine ¹

¹ University of Minnesota School of Mathematics 206 Church St. SE Minneapolis MN 55455 (USA)
² Boston College Mathematics Department 140 Commonwealth Avenue Chestnut Hill MA 02467 (USA)
³ Stanford University Department of Mathematics Building 380 Stanford California 94305 (USA)

Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 789-833.

Abstract

We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v, w$ – biaxial double $(β, q)$ -Grothendieck polynomials – which specialize at $q = 0$ and $v = 1$ to double $β$ -Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $n$ pairs of variables is a Drinfeld twist of the $U_{q} ({\hat{𝔰𝔩}}_{n + 1})$ $R$ -matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $β$ -Grothendieck polynomials and dual double $β$ -Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin–Kirillov’s Cauchy identity for $β$ -Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $β$ -Grothendieck polynomials, and prove a new branching rule for double $β$ -Grothendieck polynomials.

Received: 2021-09-30
Revised: 2022-11-14
Accepted: 2022-11-17
Published online: 2023-06-19

DOI: 10.5802/alco.277

Classification: 05E14
Keywords: Grothendieck polynomial, lattice model, Yang-Baxter equation, Cauchy identity, equivariant K-theory

Author's affiliations:

Brubaker, Ben ¹; Frechette, Claire ²; Hardt, Andrew ³; Tibor, Emily ¹; Weber, Katherine ¹

¹ University of Minnesota School of Mathematics 206 Church St. SE Minneapolis MN 55455 (USA)
² Boston College Mathematics Department 140 Commonwealth Avenue Chestnut Hill MA 02467 (USA)
³ Stanford University Department of Mathematics Building 380 Stanford California 94305 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Frozen pipes: lattice models for {Grothendieck} polynomials},
     journal = {Algebraic Combinatorics},
     pages = {789--833},
     publisher = {The Combinatorics Consortium},
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Brubaker, Ben; Frechette, Claire; Hardt, Andrew; Tibor, Emily; Weber, Katherine. Frozen pipes: lattice models for Grothendieck polynomials. Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 789-833. doi : 10.5802/alco.277. https://alco.centre-mersenne.org/articles/10.5802/alco.277/

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