On Schubert varieties of complexity one

Eunjeong Lee, Mikiya Masuda and Seonjeong Park

Vol. 315 (2021), No. 2, 419–447

DOI: 10.2140/pjm.2021.315.419

Abstract

Let $B$ be a Borel subgroup of ${GL}_n\left(ℂ\right)$ and $\mathbb{𝕋}$ a maximal torus contained in $B$. Then $\mathbb{𝕋}$ acts on ${GL}_n\left(ℂ\right)∕B$ 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 $k$. We discuss topology, geometry, and combinatorics related to Schubert varieties of complexity one.

Keywords

Schubert varieties, torus action, pattern avoidance, flag Bott–Samelson varieties, flag Bott manifolds

Mathematical Subject Classification

Primary: 14M15, 14M25

Secondary: 05A05

Milestones

Received: 8 October 2020

Revised: 16 July 2021

Accepted: 11 October 2021

Published: 19 January 2022