Vertex models for Canonical Grothendieck polynomials and their duals

Abstract

We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities. Contents 1. Introduction 1 2. Row Vertex Models 3 2.1. Definition of Physical space. 3 2.2. Row vertex model for canonical Grothendieck polynomials. 3 2.3. Row vertex model for dual canonical Grothendieck polynomials. 9 3. Column Vertex Models 3.1. Definition of Physical space. 3.2. Column vertex model for canonical Grothendieck polynomials. 3.3. Column vertex model dual canonical Grothendieck polynomials. 3.4. Vertex model for j polynomials. 4. Generalised polynomials 4.1. Difference property of the R matrices. 4.2. Generalised polynomials. 5. Duality between Column and Row models 6. Cauchy identities Appendix A. RLL relations A.1. RLL for column model of G (α,β) λ . A.2. RLL relation for row model of g (α,β) λ . A.3. RLL for column model of g (α,β) λ . A.4. RLL for the Cauchy Identity. References

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].…”

Refined dual Grothendieck polynomials, integrability, and the Schur measure

“…We remark that our lattice model is not acting on a rectangular grid, but on a "jagged" grid, which we later found this idea has only been previously used in [BBW20]. Our lattice model is also distinct from the recent work of Gunna and Zinn-Justin [GZJ20], which is partially bosonic as opposed to our completely fermionic model.…”

Determinant formulas for the five-vertex model

“…In a different crystal growth picture, it also admits an interpretation as a probabilistic cellar automaton [7]. Recent interest to the fivevertex model is motivated, in particular, by its close connection with Grothendieck polynomials [8][9][10][11][12][13]. It is also known, that in the case of a finite lattice with special fixed boundary conditions the five-vertex model configurations are in one-to-one correspondence with boxed plane partitions (3D Young diagrams which fit into a box of given size) [14,15].…”