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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. 

Abstract Let $$k \leq n$$ be positive integers, and let $$X_n = (x_1, \dots , x_n)$$ be a list of $$n$$ variables. The Boolean product polynomial$$B_{n,k}(X_n)$$ is the product of the linear forms $$\sum _{i \in S} x_i$$, where $$S$$ ranges over all $$k$$-element subsets of $$\{1, 2, \dots , n\}$$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $$B_{n,k}(X_n)$$ for certain $$k$$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $$B_{n,n-1}(X_n)$$ to a bigraded action of the symmetric group $${\mathfrak{S}}_n$$ on a divergence free quotient of superspace.