Zonotopal algebras, orbit harmonics, and Donaldson-Thomas invariants of symmetric quivers

Abstract: We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $G_{Q,\gamma}$ constructed from a symmetric quiver $Q$ with enough loops and a dimension vector $\gamma$. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson-Thomas invariants as the Hilbert series of the space of $S_{\gamma}$-invariants of the Postnikov-Shapiro slim subgraph space attached to $G_{Q,\gamma}$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical Donaldson-Thomas invariants as the number of $S_{\gamma}$-orbits under the permutation action on the set of break divisors on $G_{Q,\gamma}$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.