Let $X$ be a very general Gushel-Mukai fourfold, let $\mathcal{U}$ be the tautological sub-bundle and $\mathcal{Q}$ be the tautological quotient bundle. Let $C\subset X$ be a $\rho$-conic, then $\mathcal{U}|_C\cong\mathcal{O}_C(-1)\oplus\mathcal{O}_C(-1)$ and $\mathcal{Q}|_C\cong\mathcal{O}_C\oplus\mathcal{O}_C\oplus\mathcal{O}_C(2)$.
Now, I am trying to compute left mutation $\mathrm{L}_{\mathcal{U}}I_C$, where $I_C$ is the ideal sheaf of $C$.
get $\mathrm{RHom}(\mathcal{U},I_C)\cong k^2\oplus k[-1]$, then I would like to investigate whether the map $\pi':\mathcal{U}^2\rightarrow I_C$ is surjective. In the GM threefold case, the similar map as $\pi'$ is surjective since GM threefold does not contain any plane. I think for very general GM fourfold it is still surjective. Then we will have a short exact sequence $$0\rightarrow\mathrm{Ker}\pi'\rightarrow\mathcal{U}^2\rightarrow I_C\rightarrow 0.$$ Then $\mathrm{Ker}\pi'$ is a rank three torsion free sheaf, I guessed and tried many times, have no idea what this sheaf should be. (In dimension three case, the similar rank three sheaf is $\mathcal{Q}(-H)$, but in dimension 4, the character of $\mathrm{Ker}\pi'$ does not match that of $\mathcal{Q}(-H)$).
