Involutions in $S_n$ and associated coadjoint orbits

TL;DR

This paper investigates the coadjoint orbits of the upper triangular group related to involutions, providing formulas for their dimensions, constructing polarizations, and identifying generators of their defining ideals.

Contribution

It introduces explicit formulas and constructions for coadjoint orbits of involutions in the upper triangular group, advancing understanding of their geometric and algebraic structure.

Findings

Derived a formula for orbit dimension

Constructed a polarization for the canonical element

Identified generators of the defining ideal

Abstract

In the paper we study the coadjoint orbits of the group $\mathrm{UT}(n,K)$ associated with involutions. We obtain a formula for dimension of the orbit. We construct a polarization for the canonical element of orbit. We find a system of generators in the defining ideal of orbit.