Minimal parabolic subgroups and automorphism groups of Schubert varieties

TL;DR

This paper investigates the conditions under which certain automorphism groups of Schubert varieties are minimal parabolic subgroups, focusing on the role of minuscule fundamental weights in simple simply-laced algebraic groups.

Contribution

It characterizes minuscule fundamental weights via automorphism groups of Schubert varieties in simple simply-laced algebraic groups.

Findings

Minuscule fundamental weights correspond to specific automorphism group structures.

Automorphism groups of Schubert varieties are linked to minimal parabolic subgroups.

The paper provides criteria for automorphism groups in the context of algebraic group actions.

Abstract

Let $G$ be a simple simply-laced 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 $\omega_\alpha$ is a minuscule fundamental weight 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 $X_{Q}(w).$