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ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON SPACES OF JACOBI DIAGRAMS. II
Published online by Cambridge University Press: 09 June 2022
Abstract
The automorphism group $\operatorname {Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree d on n oriented arcs. We study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$: an action of the general linear group $\operatorname {GL}(n,\mathbb {Z})$ and an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ of the IA-automorphism group $\operatorname {IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm {gr}(\operatorname {IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$. By using this action, we study the $\operatorname {Aut}(F_n)$-module structure of $A_d(n)$. We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$-modules for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$.
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- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 1 - 69
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press
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