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.
This content will become publicly available on June 12, 2024
- Award ID(s):
- 1664858
- NSF-PAR ID:
- 10435938
- Date Published:
- Journal Name:
- Young Researcher's Forum at CG Week
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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