associative [@mkys2011 Theorem 1]. Hence, the formula has a well-defined multifactor version for products of the form $$(f_1 \in {\mathscr{H}}_{\gamma_1}) *\cdots*(f_r\in {\mathscr{H}}_{\gamma_r}) \in {\mathscr{H}}_{\sum_{u\in[r]} \gamma_u}.$$ Given the required notational complexity, we do not write the general multifactor formula explicitly here. However we do mention that the rational functions which appear in the resulting formula match the equivariant localization formula for integration along multi-step flag manifolds (instead of simply Grassmannians), a connection we exploit in Section \[s:imp.vars\].

The CoHA for $A_1$ {#ss:coha.A1}
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In this subsection we let $Q = A_1$, the quiver with a single vertex and no arrows. This example first appeared in Kontsevich–Soibelman’s seminal work [@mkys2011 Section 2.5], and is replicated in introductory sections to many papers on CoHA. We reproduce it here for completion since the tensor product factors in our main theorem (Theorem \[thm:main\]) are each subalgebras isomorphic to ${\mathscr{H}}(A_1)$.

For the quiver $A_1$, a dimension vector amounts to a choice of non-negative integer, and hence we have a decomposition ${\mathscr{H}}(A_1) = {\bigoplus}_{r\geq 0} {\mathscr{H}}_r$, where $$\label{eqn:coha.r.A1}
{\mathscr{H}}_r = {\mathrm{H}}^\bullet(B{\mathrm{GL}}({\mathbb{C}}^r)) {\cong}{\mathbb{Q}}[\omega_{1,1},\ldots,\omega_{1,r}]^{{\mathfrak{S}}_r}.$$ Throughout the remainder of the subsection, write $x_i = \omega_{1,i}$, where we warn that $x_i$ could refer to a variable in ${\mathscr{H}}_n$ for any $n\geq 