# Most primitive groups have messy invariants

• Published in 1979
In the collections
Suppose $G$ is a finite group of complex $n \times n$ matrices, and let $R^G$ be the ring of invariants of $G$: i.e., those polynomials fixed by $G$. Many authors, from Klein to the present day, have described $R^G$ by writing it as a direct sum $\sum_{j=1}^\delta \eta_j\mathrm{C}[\theta_1, \ldots, \ltheta_n]$. For example, if $G$ is a unitary group generated by reflections, $\delta = 1$. In this note we show that in general this approach is hopeless by proving that, for any $\epsilon > 0$, the smallest possible $delta$ is greater than $|G|^{n-1-\epsilon}$ for almost all primitive groups. Since for any group we can choose $\delta \leq |G|^{n-1}$, this means that most primitive groups are about as bad as they can be. The upper bound on $delta$ follows from Dade's theorem that the $\theta_i$ can be chosen to have degrees dividing $|G$.

## Other information

key
Mostprimitivegroupshavemessyinvariants
type
article
2018-05-06
date_published
1979-09-05

### BibTeX entry

@article{Mostprimitivegroupshavemessyinvariants,
key = {Mostprimitivegroupshavemessyinvariants},
type = {article},
title = {Most primitive groups have messy invariants},
author = {W.C. Huffman and N.J.A. Sloane},
abstract = {Suppose $G$ is a finite group of complex $n \times n$ matrices, and let $R^G$ be the ring of invariants of $G$: i.e., those polynomials fixed by $G$. Many authors, from Klein to the present day, have described $R^G$ by writing it as a direct sum $\sum{\_}{\{}j=1{\}}^\delta \eta{\_}j\mathrm{\{}C{\}}[\theta{\_}1, \ldots, \ltheta{\_}n]$. For example, if {\$}G{\$} is a unitary group generated by reflections, $\delta = 1$. In this note we show that in general this approach is hopeless by proving that, for any $\epsilon > 0$, the smallest possible $delta$ is greater than $|G|^{\{}n-1-\epsilon{\}}$ for almost all primitive groups. Since for any group we can choose $\delta \leq |G|^{\{}n-1{\}}$, this means that most primitive groups are about as bad as they can be. The upper bound on $delta$ follows from Dade's theorem that the $\theta{\_}i$ can be chosen to have degrees dividing $|G$.},
comment = {},
}