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 http://www-stat.stanford.edu/~cgates/PERSI/papers/toeplitz02.pdf

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?")