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

Abstract

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.Invariant theory has played a major role in 19th century mathematics. It has seen a revival in the last decades and one of the recent generalizations is noncommutative invariant theory. The study of noncommutative invariants of SL(n, C) has been initiated by Almkvist, Dicks, Formanek and Kharchenko [6,5,2], see [1] for a survey. An approach using Young tableaux was realized by Teranishi [15] and the symbolic method was adapted from the classical to the noncommutative setting by Tambour [14]. The latter provides the ground on which we establish a natural basis of the noncommutative invariants which is in bijection with certain noncrossing partitions. It arose after computer experiments and subsequent consulting of Sloane's database [12]. This bijection is applied to provide a combinatorial proof of the Molien-Weyl integral formula for the Hilbert-Poincaré series in this setting, using free cumulants and free stochastic measures.This note is organized as follows. In Section 1 we give a short survey of invariant theory and the statement of the problem. In Section 2 we review a few facts from free probability theory and noncrossing partitions. In Section 3 we explain the symbolic method and construct the noncrossing basis announced in the title. In Section 4 we review the necessary combinatorial aspects of free