explored by many authors, and have remained at the frontiers of algebraic combinatorics for two decades; see [@abwf1999; @ab2002.qv; @abakhtay2004; @ab2005.alt; @abakhtay2005; @abfsay2005; @em2005; @akemms2006; @ab2008; @rr2014; @ja2014.ir; @rkakjr2019]. In fact, the quiver polynomials are distinguished elements of the CoHA since, given a quiver orbit $\Omega \subset {\boldsymbol{\mathrm{Rep}}}_\gamma$, we have $$[\overline{\Omega}] \in {\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_\gamma}({\boldsymbol{\mathrm{Rep}}}_\gamma) = {\mathscr{H}}_\gamma.$$ For Dynkin quivers, the ${\boldsymbol{{\mathrm{GL}}}}_\gamma$-orbits in ${\boldsymbol{\mathrm{Rep}}}_\gamma$ correspond to so-called *Kostant partitions* ${{\boldsymbol{\mathrm{m}}}}{\vdash}\gamma$ (see Section \[ss:dynkins\]). Given an admissible Dynkin subquiver partition $\{Q^1,\ldots,Q^\ell\}$ for $Q$, let $\Omega_{{{\boldsymbol{\mathrm{m}}}}^j}$ denote the Dynkin quiver orbit in ${\boldsymbol{\mathrm{Rep}}}_{\gamma^j}(Q^j)$ corresponding to the Kostant partition ${{\boldsymbol{\mathrm{m}}}}^j {\vdash}\gamma^j$, where $\gamma^j$ is a dimension vector for $Q^j$, and so $\gamma = \sum_{j}\gamma^j$ is a dimension vector for $Q$. The following formula in the CoHA shows that the Dynkin quiver polynomials play a central role in the algebra structure of any acyclic CoHA.

[ **Corollary \[cor:dynkin.orbit.mult\]**.  *In terms of the isomorphism of Corollary \[cor:dynkin.decomp\], we have $$\left( \left[ \overline{\Omega_{{{\boldsymbol{\mathrm{m}}}}^1}(Q^1)} \right] \in {\mathscr{H}}_{\gamma^1}(Q^1)\right)
        * \cdots * 
        \left( \left[ \overline{\Omega_{{{\boldsymbol{\mathrm{m}}}}^\ell}(Q^\ell)} \right] \in {\mathscr{H}}_{\gamma^\ell}(Q^\ell) \right)
            = \left[ \overline{\eta}_{{\boldsymbol{\mathrm{m}}}}\right] \in {\mathscr{H}}_\gamma(Q)$$ where $\eta_{{\boldsymbol{\mathrm{m}}}}\subset {\boldsymbol{\mathrm{Rep}}}_\gamma(Q)$ is a distinguished subvariety we call a *quiver stratum* (see Section \[s:imp.vars\]).* ]{}

We now describe some historical advances for CoHA isomorphisms to put 