Littlewood–Richardson coefficients for Grothendieck polynomials from integrability

Abstract

We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant K-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-Tao and Vakil.

Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

“…And in our very recent work [30], we extended the Izergin-Korepin analysis to study the projected wavefunctions of the ( 2 ) six-vertex model. The resulting symmetric polynomials representing the projected wavefunctions contains the Grothendieck polynomials as a special case when the six-vertex model reduces to the five-vertex model [31][32][33]. We apply this technique to study the free-fermion model in an external field.…”

“…Deriving algebraic combinatorial properties of symmetric functions using their integrable model realizations is an active line of research. See [36][37][38][39][40] for more examples on Cauchy-type identities and more recent studies on the Littlewood-Richardson coefficients by [33,41].…”

Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

“…Recently, there is an active line of research which investigates relations between integrable models and related structures (integrable lattice models, classical integrable systems, vertex operators, crystal basis) and the (dual, symmetric) Grothendieck polynomials, and study the properties of the Grothendieck polynomials using the connections. See [9,10,11,12,13,14,15,16,17] for examples for various topics. We give another proof of the Guo-Sun identity (1.3) using the quantum inverse scattering method [18,19], which is a method developed to study quantum integrable models.…”

Puzzles in K-homology of Grassmannians

“…His extension works by adding a single additional tile to the set of tiles from the work of Knutson, Tao and Woodward [KTW04]. Later, Wheeler and Zinn-Justin found an alternative K-theoretic tile, that gives the structure constants of dual K-theory in an appropriate sense, see [WZ16]. Both Vakil and Wheeler-Zinn-Justin tiles have triangular shape and can be seen in Figure 2.…”