B-orbits of nilpotent order 2 and link patterns

TL;DR

This paper explores the geometric structure of nilpotent order 2 matrix orbits using link patterns, and applies these insights to orbital varieties, Springer fibers, and related combinatorial algebraic structures.

Contribution

It introduces a novel link pattern description of nilpotent order 2 orbits and uses this to analyze orbital variety closures and Springer fiber intersections.

Findings

Link patterns effectively describe orbit geometry.

Closure relations of orbital varieties are characterized.

Connections to meanders and Temperley-Lieb algebras established.

Abstract

In this paper we describe geometry of orbits of upper triangular matrices of nilpotent order 2 under conjugation by the group of upper triangular invertible matrices in terms of link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotent order 2 and intersections of components of a Springer fiber of nilpotent order 2. In particular we connect our results to the combinatorics of meanders and Temperley-Lieb algebras.