For each partition we construct a natural representation of the Lie algebra of two-variable matrix-valued polynomials. We discuss universality properties of these repreresntations as well as combinatorics of their characters. We present an explicit answer for the case of partition $(n)$, when the dimension is the higher Catalan number, as well as a simple recurrance relation in general case. At last we discuss possible generalizations for a bigger number of variables.

Parent Program: Combinatorial Representation Theory

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