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I will discuss the unreasonable effectiveness of the integrable lattice models in the theory of special functions. I will show how to use the free fermionic six vertex model to give simple, uniform proofs for most of the properties of the Schur functions that an algebraic combinatorist could think of.

Specifically, I define a new family of Schur functions which generalize and unify classical Schur functions, factorial Schur functions, supersymmetric Schur functions, Frobenius-Schur functions, factorial supersymmetric Schur functions, and dual Schur functions. I prove that the new family of functions satisfies the combinatorial description, Jacobi-Trudi identity, Nägelsbach-Kostka formula, Giambelli formula, Ribbon formula, Weyl formula, Berele-Regev factorization, and Cauchy identity.

The basis for this talk can be found in my preprint paper: https://arxiv.org/abs/2301.12110