Extensions between functors from Jacobi diagrams in handlebodies

The first Ext-groups between Schur functors in the category of modules over the $\mathbb{k}$-linearization $\mathbb{k}{\mathbf{g}\mathbf{r}}^{\mathrm{op}}$ of the opposite of the category of finitely generated free groups are computed for a filed $\mathbb{k}$ of characteristic $0$. The $\mathbb{k}$-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 $\mathbb{k}{\mathbf{g}\mathbf{r}}^{\mathrm{op}}$. We compute the first Ext-groups in the category of $\mathbf{A}$-modules between simple $\mathbf{A}$-modules which are induced by Schur functors.