Rank classification of a two-step linear map ($(r,s,t)=(\mathrm{rank}f,\mathrm{rank}g,\mathrm{rank}(gf))$)

For fixed dimensions $(d_1,d_2,d_3)$, a pair $V_1\stackrel{f}{\to }{V}_2\stackrel{g}{\to }{V}_3$ 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.