Minimal Parabolic subgroups and Automorphism groups of Schubert varieties-II

TL;DR

This paper characterizes co-minuscule roots in simple algebraic groups by examining automorphism groups of Schubert varieties within various parabolic subgroups, revealing a precise correspondence.

Contribution

It establishes a new criterion linking co-minuscule roots to automorphism groups of Schubert varieties in the context of algebraic group theory.

Findings

Co-minuscule roots correspond to specific automorphism group structures.

Automorphism groups of Schubert varieties are characterized by minimal parabolic subgroups.

The result provides a new perspective on the symmetry properties of Schubert varieties.

Abstract

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $\alpha$ is a co-minuscule root if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$