the seminal work of Buch–Fulton [@abwf1999], as well as the more recent papers [@ab2008] and [@rkakjr2019] (and references therein), as a starting point on quiver polynomials.

We observe that the quiver polynomial $[\overline{\Omega}_{{\boldsymbol{\mathrm{m}}}}]$ is an element of ${\mathscr{H}}_\gamma$, and hence an element of the CoHA. Rimányi proved that the Dynkin quiver polynomials are akin to structure constants in Dynkin CoHAs [@rr2013 Theorem 10.1]. In the sequel, we will see that these geometrically distinguished elements still play an interesting role in the algebraic structure of any acyclic CoHA.

Equivariant cohomology algebras and their elements {#ss:quivers.eq.coho}
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Fix the dimension vector $\gamma \in D$. In this paper, a central object we study is the equivariant cohomology algebra associated to the vector space and group action ; explicitly we set $$\label{eqn:defn.H.gamma}
    {\mathscr{H}}_\gamma(Q) = {\mathscr{H}}_\gamma := {\mathrm{H}}^\bullet_{{\boldsymbol{{\mathrm{GL}}}}_\gamma}({\boldsymbol{\mathrm{Rep}}}_\gamma)$$ where here and throughout the paper we assume cohomology algebras have rational coefficients. As before, we omit $Q$ from the notation when it is clear from context. Now, for each quiver vertex $i\in Q_0$, let $\{\omega_{i,j}:j\in[\gamma(i)]\}$ denote a set of indeterminates (each with cohomological degree two), which we allow to represent the Chern roots of ${\mathrm{GL}}({\mathbb{C}}^{\gamma(i)})$. Since ${\boldsymbol{\mathrm{Rep}}}_\gamma$ is a ${\boldsymbol{{\mathrm{GL}}}}_\gamma$-equivariantly contractible vector space, we further identify 