along the closed equivariant embedding $\iota: \mathrm{Rep}_{\gamma_1,\gamma_2} {\hookrightarrow}{\boldsymbol{\mathrm{Rep}}}_\gamma$, and the fourth is the pushforward along the fibration $p: BP_{\gamma_1,\gamma_2} \to B{\boldsymbol{{\mathrm{GL}}}}_\gamma$ with fiber ${\boldsymbol{{\mathrm{GL}}}}_\gamma / P_{\gamma_1,\gamma_2}$ (using that ${\boldsymbol{\mathrm{Rep}}}_\gamma$ is contractible). In particular, the fiber is a product of Grassmannians $\prod_{i\in Q_0} \mathrm{Gr}(\gamma_1(i),{\mathbb{C}}^{\gamma(i)})$.

Thus, the formula follows from the Atiyah–Bott, Berline–Vergne localization formula ([@marb1984 (3.8)], [@nbmv1982]) for equivariant pushforward mappings where $p_*$ is integration on a Grassmannian at each vertex. It is for this reason that we follow the terminology of [@rr2013] and call the formula the *equivariant localization formula* for the CoHA multiplication. The shifts in cohomological degree are $$\begin{aligned}
    c_1 & = \dim_{{\mathbb{C}}}({\boldsymbol{\mathrm{Rep}}}_{\gamma}) - \dim_{{\mathbb{C}}}(\mathrm{Rep}_{\gamma_1,\gamma_2}) &
    c_2 & = \dim_{{\mathbb{C}}}({\boldsymbol{{\mathrm{GL}}}}_\gamma/P_{\gamma_1,\gamma_2}).\end{aligned}$$ From this we see that while the $*$-multiplication respects the dimension vector grading of ${\mathscr{H}}$, its relation to cohomological degree grading is $${\mathrm{H}}^{b_1}_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma_1}}({\boldsymbol{\mathrm{Rep}}}_{\gamma_2}) 
    * {\mathrm{H}}^{b_2}_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma_2}}({\boldsymbol{\mathrm{Rep}}}_{\gamma_2})
    \subseteq {\mathrm{H}}^{b_1+b_2 - 2 \chi(\gamma_1,\gamma_2)}_{{\boldsymbol{{\mathrm{GL}}}}_{\gamma}}({\boldsymbol{\mathrm{Rep}}}_{\gamma})$$ where one checks that $c_2 = \sum_{i\in Q_0} \gamma_1(i)\gamma_2(i)$ and $c_1 = \sum_{a\in Q_1} \gamma_1(ta)\gamma_2(ha)$ so that $c_2-c_1 = \chi(\gamma_1,\gamma_2)$; e.g., compare this with the cohomological degree shifts in Examples \[ex:coha.mult\] and \[ex:coha.mult.again\].

Multifactor products in the CoHA {#ss:multifactor.products}
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A remarkable achievement of the seminal Kontsevich–Soibelman paper on CoHAs is the fact that the above multiplication is 