nLab Dickson invariant

Definition

Let $q$ be a prime power, $V$ a vector space of dimension $n$ over the finite field $\mathbf{F}_q$, and $\mathbf {G L}(V)$, the general linear group of invertible linear transformations on $V$.

The ring of Dickson invariants is defined as the $GL(V)$-invariants $Sym(V)^{\mathbf{G L}(V)}$ in the symmetric algebra on $V$.

This is a graded polynomial algebra on $n$ variables. The degrees of the generators are $q^n - q^i$ for $i=0,\dots,n-1$.

See Wilkerson 83.

Properties

Spaces with DI cohomology

The infinity-groups whose classifying spaces/deloopings have mod 2 ordinary cohomology given by rank $n$ Dickson invariants are precisely these 4 of which the first three are compact Lie groups and the last one in a 2-compact group:

$n=$	1	2	3	4
$DI(n)=$	Z/2	SO(3)	G2	G3

(Dwyer-Wilkerson 93, top of p. 38 (2 of 28))

References

Named after Leonard Dickson.

• Clarence Wilkerson, A primer on Dickson invariants, in Haynes Miller, Stewart Priddy (eds.) Proceedings of the Northwestern Homotopy Theory Conference, 1983 (doi:10.1090/conm/019) (pdf)

• William Dwyer, Clarence Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37-64 (doi:10.1090/S0894-0347-1993-1161306-9)