  We introduce a new family of Schur functions $s_{\lambda/\mu;a,b}(x/y)$ that
depend on two sets of variables and two sequences of parameters. These free
fermionic Schur functions have a hidden symmetry between the two sets of
parameters that allows us to generalize and unify factorial, supersymmetric,
and dual Schur functions from literature.
  We then prove that these functions satisfy the supersymmetric Cauchy identity
$$
  \sum_{\lambda}s_{\lambda;a,b}(x/y)\widehat{s}_{\lambda;a,b}(z/w) =
\prod_{i,j}\frac{1+y_iz_j}{1-x_iz_j}\frac{1+x_iw_j}{1-y_iw_j}, $$ where
$\widehat{s}_{\lambda;a,b}(z/w) = s_{\lambda';b',a'}(w/z)$ are the dual
functions.
  Our approach is based on the integrable six vertex model with free fermionic
Boltzmann weights. We show that these weights satisfy the \textit{refined
Yang-Baxter equation}, which allows us to prove well-known properties of Schur
functions: supersymmetry, combinatorial descriptions, the Jacobi-Trudi
identity, the N\"agelsbach-Kostka formula, the Giambelli formula, the Ribbon
formula, the Weyl determinant formula, the Berele-Regev factorization, dual
Cauchy identity, the flagged determinant formula, and many others. We emphasize
that many of these results are novel even in special cases.
