The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties

TL;DR

This paper explores the deep geometric and algebraic structures of zonotopal algebras, establishing dualities, proving conjectures, and connecting them to Schubert varieties and configuration space cohomology.

Contribution

It constructs a perfect pairing between internal zonotopal and Orlik--Terao algebras, proves a conjecture relating graph-based Orlik--Terao algebras to configuration space cohomology, and interprets zonotopal algebra inverse systems via sheaves on Schubert varieties.

Findings

Established a perfect pairing between internal zonotopal and reduced Orlik--Terao algebras.

Proved a conjecture linking graph Orlik--Terao algebra to configuration space cohomology.

Connected Macaulay inverse systems of zonotopal algebras to sheaves on Schubert varieties.

Abstract

Zonotopal algebras, introduced by Postnikov--Shapiro--Shapiro, Ardila--Postnikov, and Holtz--Ron, show up in many different contexts, including approximation theory, representation theory, Donaldson--Thomas theory, and hypertoric geometry. In the first half of this paper, we construct a perfect pairing between the internal zonotopal algebra of a linear space and the reduced Orlik--Terao algebra of the Gale dual linear space. As an application, we prove a conjecture of Moseley--Proudfoot--Young that relates the reduced Orlik--Terao algebra of a graph to the cohomology of a certain configuration space. In the second half of the paper, we interpret the Macaulay inverse system of a zonotopal algebra as the space of sections of a sheaf on the Schubert variety of a linear space. As an application of this, we use an equivariant resolution of the structure sheaf of the Schubert variety inside of a product of projective lines to produce an exact sequence relating internal and external zonotopal algebras.