A noncrossing basis for noncommutative invariants of SL(2,C)

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2,C) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory of free stochastic measures this provides a combinatorial proof of the Molien–Weyl formula in this setting.