Richard P. Stanley Seminar in Combinatorics: Structure constants for free-fermionic six-vertex model symmetric functions

The free-fermionic six-vertex model functions $F_\lambda$ and $G_\lambda$ of Aggarwal, Borodin, Petrov, and Wheeler carry inhomogeneous row and column parameters and specialize to ordinary, factorial, and supersymmetric Schur functions. We study their product expansions and show that the structure constants of $F_\kappa G_{\lambda/\mu}$, and of the product $\widetilde F_\kappa \widetilde F_{\lambda/\mu}$ of the symmetric normalizations of $F$, are partition functions of a two-color puzzle model. The puzzle weights come from a hybrid fusion of the fundamental $U_q(\widehat{\mathfrak{sl}}(1|2))$ $R$-matrix and satisfy a Yang–Baxter equation, which we apply on an auxiliary cube evaluated two ways. In the Schur limit the puzzles recover the classical Littlewood–Richardson coefficients. Joint work in progress with Michael Wheeler.

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