Nilpotent representations of the Jordan quiver

The Jordan quiver has one vertex and one loop, labelled $t$:

A finite-dimensional representation over a field $k$ is a pair $(V,x)$, where $V$ is a finite-dimensional $k$-vector space and $x:V\to V$ is nilpotent (Nilpotent endomorphisms and their nilpotency index). A morphism $(V,x)\to (V^{\mathrm{\prime }},x^{\mathrm{\prime }})$ is a linear map $f:V\to V^{\mathrm{\prime }}$ satisfying $fx=x^{\mathrm{\prime }}f$, encoded by the commutative square

For this page, ${\mathcal{J}}_k$ denotes the small model whose objects are $(k^n,X)$, for $n\ge 0$, with $X$ a nilpotent matrix, and whose morphisms are all matrices intertwining the endomorphisms. Every finite- dimensional vector space is isomorphic to a standard $k^n$, so the model represents all finite-dimensional nilpotent representations. For any field $k$, write $S=(k,0)$ for the one-dimensional object and $I_m=(k^m,J_m(0))$ for the length-$m$ Jordan block, using Jordan blocks, Jordan strings, and their endpoints. For a partition $\lambda =({\lambda }_1,\dots ,{\lambda }_s)$ (Partitions, English diagrams, and conjugation), write $I_{\lambda }$ for the block-diagonal representation with blocks $J_{{\lambda }_1}(0),\dots ,J_{{\lambda }_s}(0)$. Use $S^{\oplus 0}=I_{\mathrm{\varnothing }}=0$ for the zero object.