# On the S_n-module structure of the noncommutative harmonics

## TL;DR

This paper investigates the structure of noncommutative harmonics under the symmetric group action, deriving Frobenius series for these modules and related algebraic structures using a noncommutative Chevalley decomposition.

## Contribution

It introduces a noncommutative analog of Chevalley's decomposition to compute Frobenius series for noncommutative harmonics and related algebraic objects.

## Key findings

- Computed graded Frobenius series for noncommutative harmonics
- Derived Frobenius series for the enveloping algebra of the derived free Lie algebra
- Extended classical symmetric polynomial decompositions to noncommutative setting

## Abstract

Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.