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?

$\endgroup$

2 Answers 2

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.

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.