The geometry of zonotopal algebras I: cohomology of graphical configuration spaces
Colin Crowley, Galen Dorpalen-Barry, André Henriques, Nicholas Proudfoot
TL;DR
The paper resolves a fundamental link between combinatorial zonotopal algebras IZ(χ;ℚ) for totally unimodular vector arrangements and the cohomology of graphical configuration spaces X(,Γ) by proving IZ(Γ;ℤ) is canonically isomorphic to H^*(X(SU(2),Γ);ℤ) with a degree-halving correspondence. It introduces an orbit-harmonics interpretation, establishing IZ(χ;ℚ) ≅ gr R(χ;ℚ) and an integral form IZ(χ;ℤ) that matches integral cohomology via a Rees-graded isomorphism H^*_T(X(SU(2),Γ);ℤ) ≅ Rees R(Γ;ℤ). The results also connect to reduced Orlik–Terao algebras through duality with IZ(χ^!;ℚ), resolving aspects of the MPY conjecture canonically and providing explicit, cycle-wise and general-graph computations. By developing two filtrations, performing deletion/contraction arguments, and translating between operatic, topological, and lattice-point viewpoints, the work establishes a rigorous integral framework for these interactions and demonstrates the divided-powers structure in integral cohomology with potential implications for representation theory and combinatorial topology.
Abstract
Zonotopal algebras of vector arrangements are combinatorially-defined algebras with connections to approximation theory, introduced by Holtz and Ron and independently by Ardila and Postnikov. We show that the internal zonotopal algebra of a cographical vector arrangement is isomorphic to the cohomology ring of a certain configuration space introduced by Moseley, Proudfoot, and Young. We also study an integral form of this algebra, which in the cographical case is isomorphic to the integral cohomology ring. Our results rely on interpreting the internal zonotopal algebra of a totally unimodular arrangement as an orbit harmonics ring, that is, as the associated graded of the ring of functions on a finite set of lattice points.