Free Fermionic Schur Functions

Abstract

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.