0
$\begingroup$

Let $G$ be a finite group and let $H_G$ denote the $G$-harmonic polynomials. What is the structure of $H_G$ as a $G$-module? Is it isomorphic to the regular representation?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
2
$\begingroup$

In fact $H_G$ is isomorphic to the regular representation if and only if G is generated by (pseudo)-reflections. This is a theorem of Steinberg. A good reference for this is the manuscript "Orbit Harmonics and Graded Representations" by Haiman and Garsia.

Improve this answer
$\endgroup$
0
$\begingroup$

If $G$ is a finite reflection group, this is correct.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .