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

Colin Crowley, Nicholas Proudfoot

TL;DR

This work develops canonical dualities between zonotopal algebras (internal, central, external) and Orlik--Terao algebras via Gale duality, and recasts Macaulay inverse systems as spaces of sections on a Schubert variety $Y_{\mathcal{A}}$, enabling an $\operatorname{Aut}(\mathcal{A})$-equivariant resolution that ties internal and external algebras through a geometrically driven exact sequence. The central result identifies the internal zonotopal algebra with the dual of the reduced Orlik--Terao algebra of the Gale dual, with a concrete graph-theoretic corollary (MPY) linking graphical Ot algebras to configuration-space cohomology. The Schubert-geometry perspective yields a precise identification $H^0(\mathcal{O}_{Y_{\mathcal{A}}}(D_k)) = C_{\mathcal{A},k}$ for a range of $k$, and the equivariant resolution provides a categorified account of spanning-set identities via Möbius inversion. Together, these results weave together combinatorics, matroid theory, and Schubert geometry to produce canonical dualities and explicit syzygy descriptions of zonotopal algebras.

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.

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

TL;DR

This work develops canonical dualities between zonotopal algebras (internal, central, external) and Orlik--Terao algebras via Gale duality, and recasts Macaulay inverse systems as spaces of sections on a Schubert variety , enabling an -equivariant resolution that ties internal and external algebras through a geometrically driven exact sequence. The central result identifies the internal zonotopal algebra with the dual of the reduced Orlik--Terao algebra of the Gale dual, with a concrete graph-theoretic corollary (MPY) linking graphical Ot algebras to configuration-space cohomology. The Schubert-geometry perspective yields a precise identification for a range of , and the equivariant resolution provides a categorified account of spanning-set identities via Möbius inversion. Together, these results weave together combinatorics, matroid theory, and Schubert geometry to produce canonical dualities and explicit syzygy descriptions of zonotopal algebras.

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.
Paper Structure (21 sections, 24 theorems, 96 equations)

This paper contains 21 sections, 24 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Orlik--Terao algebra
  4. An application to graphs
  5. The Schubert variety
  6. Syzygies
  7. Modules over $\operatorname{Sym} L^\perp$
  8. OT and $\operatorname{SR}_<$
  9. $\mathcal{Q}_-$
  10. Sheaves on the poset of flats
  11. Minimal extension sheaves
  12. OT and $\operatorname{SR}_<$
  13. $\mathcal{Q}_-$
  14. Isomorphisms
  15. An example
  16. ...and 6 more sections

Key Result

Theorem 1.5

We have $\operatorname{Sym} L^* = \mathcal{P}(\mathcal{A}) \oplus \mathcal{J}(\mathcal{A}^!)$. Equivalently, the compositions are isomorphisms, and the graded vector spaces $\mathcal{P}(\mathcal{A})\cong\overline{\operatorname{SR}}(\mathcal{A}^!)$ are dual to $\mathcal{D}(\mathcal{A}^!)\cong\mathcal{R}(\mathcal{A})$.

Theorems & Definitions (61)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 51 more