Equivariant D-modules on 2x2xn hypermatrices

TL;DR

This paper investigates D-modules on the space of 2x2xn hypermatrices, classifying simple modules, analyzing their invariants, and computing cohomological properties, revealing uniform behavior for large n.

Contribution

It provides a comprehensive classification of equivariant D-modules on hypermatrix spaces and explicitly computes their invariants, extending understanding of their geometric and algebraic structure.

Findings

Classification of simple equivariant D-modules

Explicit calculation of Lyubeznik numbers and intersection cohomology

Identification of singularities and non-normal orbit closures

Abstract

We study D-modules and related invariants on the space of 2 x 2 x n hypermatrices for n >= 3, which has finitely many orbits under the action of G = GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant D-modules as the category of representations of a quiver with relations. We classify the simple equivariant D-modules, determine their characteristic cycles and find special representations that appear in their G-structures. We determine the explicit D-module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen--Macaulay. While our results display special behavior in the cases n=3 and n=4, they are completely uniform for n >= 5.