definition for CoHA multiplication {#ss:geom.coha.mult}
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The fact that produces an honest polynomial (not a rational function) can be realized as a consequence of geometry. We will make only implicit use of the geometric definition of the CoHA multiplication in the sequel, but mention some relevant details here from [@mkys2011 Section 2.2].

First, denote $\gamma = \gamma_1 + \gamma_2$. Set $P_{\gamma_1,\gamma_2}$ to be the parabolic subgroup of ${\boldsymbol{{\mathrm{GL}}}}_{\gamma}$ which preserves, for each vertex $i\in Q_0$, the subspace ${\mathbb{C}}^{\gamma_1(i)}\subseteq {\mathbb{C}}^{\gamma(i)}$. That is, in the standard basis at each vertex, $P_{\gamma_1,\gamma_2}$ consists of upper block triangular matrices that, for each vertex factor, have diagonal blocks of sizes $\gamma_1(i)$ and $\gamma_2(i)$ for each $i\in Q_0$. Moreover, we let $\mathrm{Rep}_{\gamma_1,\gamma_2}$ denote the subspace of ${\boldsymbol{\mathrm{Rep}}}_{\gamma}$ consisting of quiver representations which, for each arrow $a\in Q_1$, send the subspace ${\mathbb{C}}^{\gamma_1(ta)} \subseteq {\mathbb{C}}^{\gamma(ta)}$ into the subspace ${\mathbb{C}}^{\gamma_1(ha)} \subseteq {\mathbb{C}}^{\gamma(ha)}$. The multiplication mapping ${\mathscr{H}}_{\gamma_1} {\otimes}{\mathscr{H}}_{\gamma_2} \to {\mathscr{H}}_{\gamma}$ is defined to be the result of the following compositions [@mkys2011 Section 2.2] $$\begin{gathered}
    \label{eqn:geom.coha.mult}
{\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma_1}}({\boldsymbol{\mathrm{Rep}}}_{\gamma_1}){\otimes}{\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma_2}}({\boldsymbol{\mathrm{Rep}}}_{\gamma_2}) 
    \stackrel{{\cong}}{\longrightarrow} 
    {\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma_1} \times {\boldsymbol{{\mathrm{GL}}}}_{\gamma_2}}({\boldsymbol{\mathrm{Rep}}}_{\gamma_1}{\oplus}{\boldsymbol{\mathrm{Rep}}}_{\gamma_2}) \\
    \stackrel{{\cong}}{\longrightarrow} {\mathrm{H}}^\bullet_{P_{\gamma_1,\gamma_2}}(\mathrm{Rep}_{\gamma_1,\gamma_2})
    \stackrel{\iota_*}{\longrightarrow} {\mathrm{H}}^{\bullet+2c_1}_{P_{\gamma_1,\gamma_2}}({\boldsymbol{\mathrm{Rep}}}_{\gamma})
    \stackrel{p_*}{\longrightarrow} {\mathrm{H}}^{\bullet+2c_1-2c_2}_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma}}({\boldsymbol{\mathrm{Rep}}}_{\gamma}). \nonumber\end{gathered}$$ The first isomorphism is the Künneth formula and the second isomorphism follows from homotopy equivalence. The third map is the pushforward mapping 