Normality and non-normality of group compactifications in simple projective spaces

TL;DR

This paper characterizes when certain group compactifications in projective spaces are normal or smooth, based on the support of the dominant weight, providing precise algebraic conditions for these geometric properties.

Contribution

It offers necessary and sufficient conditions on the support of the dominant weight for the compactification to be normal or smooth, advancing understanding of their geometric structure.

Findings

Criteria for normality of $X___$ in terms of weight support

Criteria for smoothness of $X___$ in terms of weight support

Complete characterization of geometric properties based on algebraic data

Abstract

If $G$ is a complex simply connected semisimple algebraic group and if $\lambda$ is a dominant weight, we consider the compactification $X_\lambda$ in the projectivisation of $\End(V(\lambda))$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of $\lambda$ so that $X_\lambda$ is normal; as well, we give necessary and sufficient conditions on the support of $\lambda$ so that $X_\lambda$ is smooth.