Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{S} \subseteq \mathcal{T}$ be an admissible subcategory. Recall, this means that the inclusion $\mathcal{S} \hookrightarrow \mathcal{T}$ admits both left and right adjoints. For the sake of discussion, let us consider the right adjoint $\tau : \mathcal{T} \rightarrow \mathcal{S}$.
My question is whether there are examples where the functor $\tau : \mathcal{T} \rightarrow \mathcal{S}$ can be computed?
For example, the composition $\mathcal{S} \hookrightarrow \mathcal{T} \rightarrow \mathcal{S}$ is the identity, and you can try to understand homs by considering the exact triangle
$\tau(A) \rightarrow A \rightarrow B$,
where $B \in S^{\perp}$.
However, I have not been able to find examples where the functor $\tau : \mathcal{T} \rightarrow \mathcal{S}$ can be explicitly computed.
I am particularly interested in the case that $S$ is generated by an exceptional collection, and is a subcategory of the bounded derived category of perfect complexes on the grassmaninan $D^{b}_{perf}(Gr_{k,n})$. (Grassmanian taken over a field of characteristic zero).
In the case I am considering, I know that ${}^{\perp}S$ and $S^{\perp}$ are generated by exceptional objects, and the functor $\tau_{S^{\perp}}: {}^{\perp}S \rightarrow S^{\perp}$ is well-defined and an equivalence. What I am trying to do is compute this functor to show it maps the exceptional objects in the way that I think it does, which is where I am getting stuck.
Any help/ references to known computations would be much appreciated!
