Zonotopal Algebras, Orbit Harmonics, and Donaldson–Thomas Invariants of Symmetric Quivers

Abstract

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 (DT) 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 DT invariants as the number of $S_{\gamma }$-orbits under the permutation action on the set of break divisors on $G$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.