  We examine and present new combinatorics for the Schur polynomials from the
viewpoint of quantum integrability. We introduce and analyze an integrable
six-vertex model which can be viewed as a certain degeneration model from a
t-deformed boson model. By a detailed analysis of the wavefunction from the
quantum inverse scattering method, we present a novel combinatorial formula
which expresses the Schur polynomials by using an additional parameter, which
is in the same sense but different from the Tokuyama formula. We also give an
algebraic analytic proof for the Cauchy identity and make applications of the
domain wall boundary partition functions to the enumeration of alternating sign
matrices.
