Interesting Esoterica

Most primitive groups have messy invariants

Article by W.C. Huffman and N.J.A. Sloane
  • Published in 1979
  • Added on
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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 \(|G\).

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BibTeX entry

@article{Mostprimitivegroupshavemessyinvariants,
	title = {Most primitive groups have messy invariants},
	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 \(|G\).},
	url = {https://www.sciencedirect.com/science/article/pii/0001870879900380 http://neilsloane.com/doc/Me61.pdf},
	year = 1979,
	author = {W.C. Huffman and N.J.A. Sloane},
	comment = {},
	urldate = {2018-05-06},
	collections = {Attention-grabbing titles}
}