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

TL;DR

This paper introduces a noncrossing basis for noncommutative invariants of SL(2,C), providing a combinatorial explanation and proof of the invariant space dimensions using noncrossing partitions and free stochastic measures.

Contribution

It constructs a noncrossing basis for invariant spaces and offers a combinatorial proof of the Molien-Weyl formula in noncommutative invariant theory.

Findings

Dimensions of invariant spaces match counts of noncrossing partitions

Constructed a noncrossing basis for homogeneous components

Provided a combinatorial proof of the Molien-Weyl formula

Abstract

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of $SL(2,\IC)$ 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 free stochastic measures this provides a combinatorial proof of the Molien-Weyl formula in this setting.