Normality of closure of orthogonal nilpotent symmetric orbits

TL;DR

This paper investigates the normality of closures of orthogonal nilpotent symmetric orbits, revealing that only a small subset have normal closures, contrasting with the symplectic case, through combinatorial methods.

Contribution

It identifies which orthogonal nilpotent symmetric orbit closures are normal, extending the understanding of orbit closure properties in symmetric matrices.

Findings

Only a small portion of classes have normal closure

Contrast with symplectic group where all classes are normal

Uses combinatorial computations inspired by Kraft-Procesi and Ohta

Abstract

We study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.