Local cohomology with support in ideals of symmetric minors and Pfaffians

TL;DR

This paper computes local cohomology modules supported on determinantal varieties of symmetric and skew-symmetric matrices, describing their structure and multiplicities, and deriving formulas for cohomological dimensions related to minors.

Contribution

It provides explicit descriptions of local cohomology modules and their composition factors for symmetric and skew-symmetric matrices, extending previous results to new cases.

Findings

Explicit formulas for local cohomology modules and their multiplicities.

A formula for the cohomological dimension of ideals of even minors.

Decomposition of local cohomology modules into irreducible GL-representations.

Abstract

We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the case of odd minors, this was obtained by Barile in the 90s. Another consequence of our work is that we obtain a description of the decomposition into irreducible GL-representations of the local cohomology modules (the analogous problem in the case when X is the vector space of m x n matrices was treated in earlier work of the authors).