Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice. (English) Zbl 1375.81131
Summary: We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in a finite lattice. We introduce the \(L\)-operator related to the \(\operatorname{U}_q(\operatorname{sl}_{2})\; R\)-matrix and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the \(B\)-operators.{
©2017 American Institute of Physics}
©2017 American Institute of Physics}
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 16T25 | Yang-Baxter equations |
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
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