Characteristic classes of finite group representations

[wofsy]

I know zero about the characteristic classes of finite group representations and would appreciate a reference.

specifically, if I have a faithful representations of a finite group,G, in O(n) what can I say about the induced map on cohomology,

P**(BO(n))-> H*(BG) ?

I am mostly interested in Z or Z/2 coefficients so this would be Pontryagin classes and Stiefel-Whitney classes.

For instance what are the induced maps of these two representations of Z/2?

-1 reflection of the real line

-1 0
0 -1 180 degree rotation of R^2.Can you show me an unoriented representation of the dihedral group of order 8 so that the induced map on Z/2 cohomology is not surjective?

 

[jvicens]

Sure, I would be happy to provide some references for you to learn more about characteristic classes of finite group representations.

Firstly, a great starting point would be the book "Representation Theory: A First Course" by William Fulton and Joe Harris. This book covers the basics of representation theory and includes a section on characteristic classes of finite group representations.

Another helpful resource would be the article "Characteristic Classes of Finite Group Representations" by Allen Knutson, which can be found on the arXiv website.

In terms of your specific question about the induced map on cohomology, there is a theorem known as the "Borel-Weil-Bott Theorem" which gives a formula for the Chern classes of a representation in terms of the weights of the representation. This can be used to calculate the induced map on cohomology.

For the specific examples you mentioned, the induced map on Z/2 cohomology for the representation of Z/2 by -1 (reflection of the real line) would be the identity map, since the real line has no non-trivial cohomology. For the representation of Z/2 by the 180 degree rotation of R^2, the induced map on Z/2 cohomology would be the map that sends the generator of H^2(BO(2)) to the generator of H^2(BZ/2)).

As for an unoriented representation of the dihedral group of order 8 where the induced map on Z/2 cohomology is not surjective, one example would be the standard representation of D_8 on R^2, where the rotation by 180 degrees acts as -1 on R^2. In this case, the induced map on Z/2 cohomology would not be surjective since the generator of H^2(BZ/2) is not in the image of the induced map.

I hope this helps and provides a good starting point for your further exploration of characteristic classes of finite group representations. Good luck!