= Rank classification of a two-step linear map
{title2=$(r,s,t)=(\operatorname{rank}f,\operatorname{rank}g,\operatorname{rank}(gf))$}
For fixed dimensions $(d_1,d_2,d_3)$, a pair $V_1\xrightarrow fV_2\xrightarrow gV_3$ is classified up to vertexwise change of basis by its three ranks $r,s,t$. The multiplicities of its <interval representations of an equioriented three-vertex quiver> are $m_{13}=t$, $m_{12}=r-t$, $m_{23}=s-t$, $m_{11}=d_1-r$, $m_{22}=d_2-r-s+t$ and $m_{33}=d_3-s$. Nonnegativity is precisely the existence condition. For dimensions $(2,2,2)$ there are ten orbits, of which $(2,2,2)$ is open; its boundary has nine orbits. The two cases $(1,1,0)$ and $(1,1,1)$ show why the individual arrow ranks are insufficient.
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