${\mathscr{H}}$ is generated. Our main results provide one possible point of view on the CoHA isomorphism problem; it is our hope that readers familiar with characteristic classes of singular varieties (specifically the story of quiver polynomials) will find our description of the CoHA approachable. For example, we state one of our main results below.

[ **Corollary \[cor:dynkin.decomp\]**.  *For certain choices of Dynkin subquivers $Q^1,\ldots,Q^\ell$ of $Q$ (we call such a choice an *admissible Dynkin subquiver partition*) the (left to right) $*$-multiplication induces an isomorphism $$\pushQED{\qed}{\mathscr{H}}(Q^1) {\otimes}\cdots {\otimes}{\mathscr{H}}(Q^\ell) \stackrel{{\cong}}{\longrightarrow} {\mathscr{H}}(Q). \qedhere\popQED$$* ]{}

As in the statement above, the main results in this paper live in a family parameterized by a choice of an *admissible Dynkin subquiver partition* of $Q$ [@ja2018]. The seminal work of Gabriel [@pg1972] established the *Dynkin quivers* (orientations of simply-laced Dynkin diagrams of type $A$, $D$, or $E$) as the “base case” for considering the representations of acyclic quivers. An important chapter in the study of Dynkin quiver representations is the theory of *quiver polynomials*. These are characteristic fundamental classes associated to the closure (in ${\boldsymbol{\mathrm{Rep}}}_\gamma$) of a ${\boldsymbol{{\mathrm{GL}}}}_\gamma$-orbit (equivalently, the locus of quiver representations of a fixed isomorphism type). Their remarkable positivity and stability properties have been 