Projective normality of Weyl group quotients

TL;DR

This paper proves that quotients of projective spaces by Weyl groups of certain types are projectively normal, extending to any finite group and representation, with implications for algebraic geometry and invariant theory.

Contribution

It establishes projective normality of Weyl group quotients for classical types and general finite groups, generalizing previous results in invariant theory.

Findings

Weyl group quotients of classical types are projectively normal.

Finite group quotients are projectively normal with respect to specific line bundle descent.

Results apply to a broad class of finite groups and representations.

Abstract

In this note, we prove that for the standard representation $V$of the Weyl group $W$ of a semi-simple algebraic group of type $A_n, B_n, C_n, D_n, F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb C$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes n!}$ as a consequence.