Unitary matrix models, quantized symmetric functions and spin chain

We construct a correspondence between a broad class of unitary matrix models and vacuum correlation functions in quantum spin-chain Hilbert spaces. The key step is to lift symmetric functions to operators acting on the $N$-magnon sector in a way that preserves the relevant ring structure. For any unitary matrix model whose integrand admits a factorized expansion in symmetric functions, the Schur orthogonality of the unitary group integral is then translated into the inner product of quantized Schur states. We illustrate the construction for the Gross--Witten--Wadia model, superconformal indices of $\mathcal{N}=4$ super Yang--Mills theory with classical gauge groups, and Toda tau functions. The resulting operator formulation provides a unified algebraic bridge between unitary matrix integrals, quantum integrable systems and symmetric function theory.