Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extensions between functors from Jacobi diagrams in handlebodies

Mai Katada

TL;DR

This work determines the first extension groups between simple modules in the category of Jacobi diagrams in handlebodies by exploiting an analytic equivalence with a Casimir Lie algebra PROP. The core method transfers Ext calculations to the well-structured $ extbf{Cat}_{ ext{Lie}^{ ext{C}}}$-Mod setting, yielding explicit descriptions: $ ext{Ext}^1$ vanishes unless the row/column sizes satisfy $m=n-1$ or $m=n+2$, with dimensions governed by Littlewood–Richardson coefficients and the upward Brauer category, and then transfers back to $ extbf{A}$-Mod$^{ ext{ω}}$ (and directly to $ extbf{A}$-Mod) via established adjunctions and equivalences. The paper also directly analyzes extensions for symmetric and exterior power functors, providing concrete 1-dimensional spaces in the $d' eq d+2$ case and explicit nontrivial extensions when $d'=d+2$, together with a representation-theoretic realization in terms of Jacobi-diagram modules. Overall, the results connect cohomological data of functor categories to classical symmetric-group representation theory and Casimir-Lie structures, with potential implications for the cohomology of automorphism groups of free groups and related topological invariants.

Abstract

The first Ext-groups between Schur functors in the category of modules over the $\Bbbk$-linearization $\Bbbk\mathbf{gr}^{\operatorname{op}}$ of the opposite of the category of finitely generated free groups are computed for a filed $\Bbbk$ of characteristic $0$. The $\Bbbk$-linear category $\mathbf{A}$ of Jacobi diagrams in handlebodies, which was introduced by Habiro and Massuyeau, has an $\mathbb{N}$-grading whose degree $0$ part identifies with the category $\Bbbk\mathbf{gr}^{\operatorname{op}}$. We compute the first Ext-groups in the category of $\mathbf{A}$-modules between simple $\mathbf{A}$-modules which are induced by Schur functors.

Extensions between functors from Jacobi diagrams in handlebodies

TL;DR

This work determines the first extension groups between simple modules in the category of Jacobi diagrams in handlebodies by exploiting an analytic equivalence with a Casimir Lie algebra PROP. The core method transfers Ext calculations to the well-structured -Mod setting, yielding explicit descriptions: vanishes unless the row/column sizes satisfy or , with dimensions governed by Littlewood–Richardson coefficients and the upward Brauer category, and then transfers back to -Mod (and directly to -Mod) via established adjunctions and equivalences. The paper also directly analyzes extensions for symmetric and exterior power functors, providing concrete 1-dimensional spaces in the case and explicit nontrivial extensions when , together with a representation-theoretic realization in terms of Jacobi-diagram modules. Overall, the results connect cohomological data of functor categories to classical symmetric-group representation theory and Casimir-Lie structures, with potential implications for the cohomology of automorphism groups of free groups and related topological invariants.

Abstract

The first Ext-groups between Schur functors in the category of modules over the -linearization of the opposite of the category of finitely generated free groups are computed for a filed of characteristic . The -linear category of Jacobi diagrams in handlebodies, which was introduced by Habiro and Massuyeau, has an -grading whose degree part identifies with the category . We compute the first Ext-groups in the category of -modules between simple -modules which are induced by Schur functors.

Paper Structure

This paper contains 11 sections, 16 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The category $\mathbf{A}$ of Jacobi diagrams in handlebodies
  4. The $\Bbbk$-linear PROP $\mathcal{C}at_{\mathcal{L}ie^{\mathrm{C}}}$ for Casimir Lie algebras
  5. Equivalence of categories
  6. Specht modules and Schur functors
  7. The first Ext-groups in the category of $\mathbf{A}$-modules
  8. The Ext-groups in $\Bbbk\mathbf{gr}^{\operatorname{op}}\textbf{-}\mathrm{Mod}$
  9. The Yoneda Ext
  10. The first Ext-groups in $\mathcal{C}at_{\mathcal{L}ie^{\mathrm{C}}}\textbf{-}\mathrm{Mod}$
  11. The computation in $\mathbf{A}\textbf{-}\mathrm{Mod}$

Key Result

Lemma 1.1

Let $\lambda,\mu$ be partitions and set $n=|\lambda|$ and $m=|\mu|$. Then we have where $LR^{\bullet}_{\bullet,\bullet}$ denotes the Littlewood--Richardson coefficient.

Theorems & Definitions (24)

  • Lemma 1.1: Powell--Vespa, see Lemma \ref{['extgrop']}
  • Theorem 1.2: see Theorem \ref{['extcatLieC']}
  • Theorem 1.3: see Theorem \ref{['Ext1Amod']}
  • Lemma 2.1: Habiro--Massuyeau
  • Lemma 2.2: Habiro--Massuyeau
  • Proposition 2.3: Powellanalytic
  • Proposition 2.4: katadaAmod
  • Proposition 2.5: katadaAmod
  • Proposition 2.6: Kim Kimtalk
  • Lemma 3.1
  • ...and 14 more