Equivariant spaces of matrices of constant rank

TL;DR

This paper employs representation theory to construct and analyze spaces of matrices with constant rank, exploring their parametrizations, deformations, and connections to homogeneous vector bundles.

Contribution

It introduces new constructions of matrix spaces of constant rank using various group representations and studies their deformation properties.

Findings

Spaces are parametrized by general linear, symplectic, and orthogonal groups.

Some matrix spaces admit large families of deformations.

Connections established between matrix spaces and homogeneous vector bundles.

Abstract

We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more complicated representations, and others with the orthogonal group. Our spaces of matrices correspond to vector bundles which are homogeneous but sometimes admit deformations to non-homogeneous vector bundles, showing that these spaces of matrices sometimes admit large families of deformations.