A noncrossing basis for noncommutative invariants of

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.