  We apply the Izergin-Korepin analysis to the study of the projected
wavefunctions of the generalized free-fermion model. We introduce a
generalization of the $L$-operator of the six-vertex model by
Bump-Brubaker-Friedberg and Bump-McNamara-Nakasuji. We make the Izergin-Korepin
analysis to characterize the projected wavefunctions and show that they can be
expressed as a product of factors and certain symmetric functions which
generalizes the factorial Schur functions. This result can be seen as a
generalization of the Tokuyama formula for the factorial Schur functions.
