$\langle \beta',\beta''\rangle \geq 0$;

-   positive roots from *different* subquivers, then $\langle \beta',\beta''\rangle \leq 0$.

From the above, we see that this Reineke order convention on $\Phi_+({{\mathcal}{Q}^\bullet})$ is equivalent to the ordering convention on the same set in [@ja2018 Section 2.5].

Cohomological Hall algebras of quivers {#s:coha}
======================================

As a $D$-graded vector space, the *cohomological Hall algebra* (aka CoHA) *of a quiver $Q$* is $$\label{eqn:defn.CoHA}
    {\mathscr{H}}(Q) = {\mathscr{H}}:= {\bigoplus}_{\gamma\in D} {\mathscr{H}}_\gamma \nonumber$$ where we recall that ${\mathscr{H}}_\gamma$ is the equivariant cohomology algebra from and we think of the elements of ${\mathscr{H}}_\gamma$ as polynomials via .

Equivariant localization formula for CoHA multiplication {#ss:eq.local.coha.mult}
--------------------------------------------------------

For $\gamma_1,\gamma_2\in D$ and $f_1\in {\mathscr{H}}_{\gamma_1}$, $f_2\in{\mathscr{H}}_{\gamma_2}$, we obtain $f_1 * f_2 \in {\mathscr{H}}_{\gamma_1+\gamma_2}$ via the formula (see [@mkys2011 Theorem 2]) $$\begin{gathered}
        \label{eqn:coha.two.factor}
(f_1*f_2)(\omega_{1,1},\ldots,\omega_{1,\gamma_1(1)+\gamma_2(1)};\ldots;\omega_{n,1},\ldots,\omega_{n,\gamma_1(n)+\gamma_2(n)}) \\
 = \sum_{S_1 \in \binom{[\gamma_1(1) + \gamma_2(1)]}{\gamma_1(1)}} 
 \cdots 
    \sum_{S_n \in \binom{[\gamma_1(n) + \gamma_2(n)]}{\gamma_1(n)}}
        f_1(S_\bullet)f_2(\overline{S_\bullet}) \frac{\prod_{a\in Q_1} (\omega_{ha,\overline{S_{ha}}} - \omega_{ta,S_{ta}} )}{\prod_{i\in Q_0} (\omega_{i,\overline{S_i}} - \omega_{i,S_i} )}.\end{gathered}$$ To compute a term in the sum above, for each vertex $i\in Q_0$ we must choose a set $$S_i \in \binom{[\gamma_1(i) + \gamma_2(i)]}{\gamma_1(i)}$$ which is a subset of $[\gamma_1(i) + \gamma_2(i)]$ with $\gamma_1(i)$ elements, and $\overline{S_i}$ denotes its complement in $[\gamma_1(i) + \gamma_2(i)]$ (and so has $\gamma_2(i)$ elements). 