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Dickson invariant

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Definition

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

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

This is a graded polynomial algebra on nn variables. The degrees of the generators are q nq iq^n - q^i for i=0,,n1i=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 nn 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=n=1234
DI(n)=DI(n)=Z/2SO(3)G2G3

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

References

Named after Leonard Dickson.

Last revised on August 26, 2019 at 12:16:05. See the history of this page for a list of all contributions to it.