Frozen Pipes: Lattice Models for Grothendieck Polynomials

Abstract

We prove the existence of several different families of solvable lattice models whose partition functions give the double β-Grothendieck polynomials and the dual double β-Grothendieck polynomials for arbitrary permutations. Moreover, we introduce a new family of double "biaxial" β-Grothendieck polynomials depending on a pair of permutations which simultaneously generalize both the double and dual double polynomials. We then use these models and their Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for β-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial β-Grothendieck polynomials, and prove a new branching rule for double β-Grothendieck polynomials.and Wheeler [4]. Unlike Borodin-Wheeler, who use color to refine certain partition functions of lattice models from symmetric functions into their nonsymmetric pieces, our models use color to move from permutations with one descent to those with arbitrarily many descents. As a nod to this similarity, we refer to our models as "chromatic" rather than "colo(u)red." As a further distinction, the associated quantum group module for our solutions of the Yang-Baxter equation (at least in the specialization β = −1) is a Drinfeld twist of the standard U q (sl n+1 ) module, while those in [4] arise from symmetric powers of U q (sl 2 ) modules (see Remark 4.2).We conclude by outlining the structure of the subsequent sections. In Section 2, we present two different definitions for double β-Grothendieck polynomials which reflect the Hecke algebra point of view and the combinatorial generating function point of view discussed earlier. Then in Section 3, we introduce two similar, but not bijectively equivalent, lattice models whose partition functions give the double β-Grothendieck polynomials. By allowing more general boundary conditions indexed by a pair of partitions, we obtain what we call the "biaxial" model, which will turn out to simultaneously generalize both the double β-Grothendieck polynomials and their duals. In Section 4, we describe the Yang-Baxter equations and R-matrices associated to each model, and in Section 5, we use these solutions to compute the partition functions. We also briefly discuss the correspondence between the states of the models and pipe dreams. In Section 6, we prove Cauchy-type identities by stacking our models appropriately and calculating the partition functions in two ways. Finally, in Section 7, we provide a branching rule that describes how to express a double β-Grothendieck polynomial for any permutation in S n as a sum over double β-Grothendieck polynomials for permutations in S n−1 .

Refined canonical stable Grothendieck polynomials and their duals

“…Finally, as mentioned in the introduction it is known that Grothendieck polynomials G λ (x) and g λ (x) have integrable vertex models [6,8,19,33,34,41], crystal structures [14,20,35] and probabilistic models [34,43]. It would be very interesting to extend these results to G λ (x; α, β) and g λ (x; α, β).…”

“…Besides the Jacobi-Trudi formula, Grothendieck polynomials have been studied from various points of view. In particular, it is known that Grothendieck polynomials have integrable vertex models [6,8,19,33,34,41], crystal structures [14,20,35], and probabilistic models [34,43].…”

An orthodontia formula for Grothendieck polynomials

“…Many formulas for Schubert polynomials generalize to Grothendieck polynomials. Recent work [3,4,20] has uncovered novel formulas for Grothendieck polynomials and their generalizations.…”

Quasi-solvable lattice models for and Demazure atoms and characters

“…The train argument consists of adding an R-matrix to a pair of rows and then passing it through to the other side by repeated use of the Yang-Baxter equation. This is what was used to show that the functional equations satisfied the divided difference operator relations in [16,18,19,20] and the 𝑧 𝑖 ↔ 𝑧 𝑗 symmetry in [31,35,37]. To produce functional equations for U-turn lattice models as for the uncoloured model, two additional types of relations are needed.…”