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Noncommutative invariant theory is a generalization of the classicalinvariant theory of the action of $SL(2,\IC)$ on binary forms. The dimensionsof the spaces of invariant noncommutative polynomials coincide with the numbersof certain noncrossing partitions. We give an elementary combinatorialexplanation of this fact by constructing a noncrossing basis of the homogeneouscomponents. Using the theory free stochastic measures this provides acombinatorial proof of the Molien-Weyl formula in this setting.

A Noncrossing Basis for Noncommutative Invariants of SL(2,C)