Hook formulas for skew shapes via contour integrals and vertex models

The celebrated hook-length formula (HLF) of Frame-Robinson-Thrall, which gives the dimension of irreducible $S_n$ modules and the number of standard Young tableaux (SYT), has been at the heart of many results from algebraic combinatorics, representation theory and integrable probability.

No such closed formula exists for counting SYTs of skew shapes, the closest formula to it (called NHLF) emerged through implicit computations in equivariant Schubert calculus giving a hook-product weighted sum over so-called excited diagrams.  Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also nonintersecting lattice paths inside a Young diagram and the NHLF has seen a variety of applications from weighted lozenge tilings to asymptotics of skew SYTs. We give two self-contained proofs of a multivariate generalization of this formula, which allow us to extend the setup beyond standard Young tableaux and the underlying Schur polynomials. The first proof uses multiple contour integrals. The second one interprets excited diagrams as configurations of a six-vertex model at a free fermion point, and derives the formula for the number of standard Young tableaux of a skew shape from the Yang-Baxter equation.

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