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We introduce and study several combinatorial properties of a class ofsymmetric polynomials from the point of view of integrable vertex models infinite lattice. We introduce the $L$-operator related with the $U_q(sl_2)$$R$-matrix, and construct the wavefunctions and their duals. We prove the exactcorrespondence between the wavefunctions and symmetric polynomials which is aquantum group deformation of the Grothendieck polynomials. This is proved bycombining the matrix product method and an analysis on the domain wall boundarypartition functions. As applications of the correspondence between thewavefunctions and symmetric polynomials, we derive several properties of thesymmetric polynomials such as the determinant pairing formulas and thebranching formulas by analyzing the domain wall boundary partition functionsand the matrix elements of the $B$-operators.

Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice