On Schubert varieties of complexity one

TL;DR

This paper investigates the topology, geometry, and combinatorics of Schubert varieties of complexity one, which are characterized by having a maximal torus orbit of codimension one, providing insights into their structure.

Contribution

It offers a detailed analysis of Schubert varieties of complexity one, exploring their topological, geometric, and combinatorial properties, which was previously less understood.

Findings

Characterization of Schubert varieties of complexity one

Topological and geometric properties elucidated

Combinatorial structures related to these varieties analyzed

Abstract

Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a maximal torus contained in $B$. Then $\mathbb{T}$ acts on $\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is $\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.