Consider a finite group $G$ and its complex group algebra $V_G$, on which $G$ acts. I would like to know: what are the polynomial $G$-invariants of $V_G$ i.e., the polynomial functions $p\in \mathbb{C}[V_G]$ that are $G$-invariant? Thank you very much for the help.
$\begingroup$
$\endgroup$
2
-
$\begingroup$ Are you asking for the polynomial invariants of the regular action of $G$, i.e. the linear group $G$ acting on $V_G$? $\endgroup$– Dima PasechnikCommented Oct 27, 2022 at 7:57
-
$\begingroup$ Yes, $V_G$ is the regular representation of $G$. $\endgroup$– user493645Commented Oct 27, 2022 at 8:00
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
This is a particular case of rings of invariants of permutation groups. The $n=|G|$ elementary symmetric polynomials $\sigma_k$ form a system of parameters for $R:=\mathbb{C}[V_G]^G$, the ring in question, and $R$ is a free module of rank $(n-1)!$ over the subring generated by the $\sigma_k$.
See Sect. 2.7 of B.Sturmfels' "Algorithms in invariant theory", ISBN 978-3-211-77416-8, for details.
answered Oct 27, 2022 at 11:03
-
$\begingroup$ Thank you very much for the answer. What about cyclic groups or in general finite Abelian groups? Is there a known answer in this case? $\endgroup$ Commented Oct 27, 2022 at 12:25
-
1$\begingroup$ there is a discussion in the book I referred to. I don't think there is an explicit answer known. $\endgroup$ Commented Oct 27, 2022 at 19:37