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By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example

Now, my question is as follows: Is there a similar way to interpret banded block Toeplitz matrices in terms of Schur polynomials? (Maybe one needs to have some generalized version of Schur polynomials, or matrix-valued "polynomials?")

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