Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

TL;DR

This paper investigates how automorphism groups of free groups act on spaces of Jacobi diagrams, revealing their module structure and providing explicit decompositions and polynomial functor constructions.

Contribution

It introduces a detailed analysis of the automorphism group actions on Jacobi diagram spaces, including indecomposable decompositions and a polynomial functor framework.

Findings

Decomposition of $A_2(n)$ into indecomposable modules.

Construction of a polynomial functor $A_d$ capturing automorphism actions.

Analysis of the module structure of Jacobi diagram spaces.

Abstract

We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated graded vector space of $A_d(n)$, which is identified with the space $B_d(n)$ of open Jacobi diagrams, an action of the general linear group $\mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the IA-automorphism group of $F_n$ associated with its lower central series. We use these actions on $B_d(n)$ to study the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an indecomposable decomposition of $A_2(n)$. We also construct a polynomial functor $A_d$ of degree $2d$ from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $n\geq 0$.