Jim Caprioli at The Finland Station

Still working on the first paragraph...

"... A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism p: G -> GL(V) of G to the group of automorphisms of V; we say that such a map gives V the structure of a G-module. When there is little ambiguity about the map p (and, we're afraid, even sometimes when there is) we sometimes call V itself a representation of G; in this vein we will often suppress the symbol p and write g. v or gv for p(g)(v). The dimension of V is sometimes called the degree of p. ..."

In my words.

Representation
- is a homomorphism from Group to GLn(F).
- A rep. maps a group element to a square matrix with elements in field F (usually complex numbers)
- Identity element is always mapped to the identity matrix.
- Homomorphism rules apply. p(g*h) = p(g)*p(h).

CG module.
- is a group action ( and here ) on a vector space V
- f: G x V -> V: f(g,v) = p(g)*v.

Faithful CG-module
- CG module
- p(g)*v = v <=> g = identity in G.

It is possible to make this less abstract by taking
- vector spaces V=R2 or V=R3
- some geometric objects like a square or some n-gon, whatever
- take a Dihedral Group
- map the elements of the Dihedral Group to a matrix
- take some basis in R2 or R3 (not the obvious one) and calculate the matrix for a group element
- see the action 'work'
That is what it is -about-. I think.

I made -some- progress today. Not much but measurable. It feels as such. It is weeks ago since I had a feeling of progress.