The combinatorics of orbital varieties closures of nilpotent order 2 in sl(n)

TL;DR

This paper explores the relationship between two partial orders on Young tableaux and reveals their equivalence in specific cases, leading to new geometric insights into orbital varieties of nilpotent order 2 in sl(n).

Contribution

It demonstrates the coincidence of two distinct partial orders on Young tableaux with 2 rows or columns and links this to geometric properties of orbital varieties in Lie algebra.

Findings

The two partial orders coincide on tableaux with 2 rows or columns.

This coincidence has geometric implications for orbital varieties of nilpotent order 2.

Provides a new combinatorial perspective on orbital varieties in sl(n).

Abstract

We consider two partial orders on standard Young tableaux. The first one is induced from the weak right Bruhat order on symmetric group by Robinson-Schensted algorithm. The second one is induced from the order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We show that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in sl(n).