References for functor cohomology and applications:

Several aspects are treated in the following (it includes many references as well) :

MR2117525 Franjou, Vincent; Friedlander, Eric M.; Pirashvili, Teimuraz; Schwartz, Lionel Rational representations, the Steenrod algebra and functor homology. Panoramas et Synthèses [Panoramas and Syntheses], 16. Société Mathématique de France, Paris, 2003. xxii+132 pp.

Recent computations by Chalupnik and Touzé are presented (with references) in :

arXiv:math/0701504 [ps, pdf, other]

Title: Cohomologie du groupe linéaire à coefficients dans les polynômes de matrices

Authors: Antoine Touzé

MR2195259 (2006k:20090) Cha\l upnik, Marcin Extensions of strict polynomial functors. Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 773--792. (Reviewer: Anne Elisabeth Henke)

PDF Doc Del Clipboard Journal Article

Kuhn's series of paper in the 90's are nice to read:

MR1344142 (97c:55026) Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. III. $K$-Theory 9 (1995), no. 3, 273--303. (Reviewer: Haynes R. Miller)

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MR1300547 (95k:55038) Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. II. $K$-Theory 8 (1994), no. 4, 395--428. (Reviewer: Haynes R. Miller)

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MR1269607 (95c:55022) Kuhn, Nicholas J. Generic representations of the finite general linear groups and the Steenrod algebra. I. Amer. J. Math. 116 (1994), no. 2, 327--360. (Reviewer: Haynes R. Miller)

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"Strict" polynomial functors:

In the following great paper, a version of polynomial functors is build that tauntamounts to polynomial representations of Schur algebras, and it is used to prove finite generation of the cohomology for finite-dimensional cocommutative Hopf algebras (or finite group schemes):

MR1427618 (98h:14055a) Friedlander, Eric M.; Suslin, Andrei Cohomology of finite group schemes over a field. Invent. Math. 127 (1997), no. 2, 209--270. (Reviewer: Daniel K. Nakano)

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For a presentation and a generalisation of Nagata's result to higher invariants (putting the above in perspective), I like the first 6 pages of

arXiv:math/0403361 [ps, pdf, other]

Title: A reductive group with finitely generated cohomology algebras

Authors: Wilberd van der Kallen

Invariant theory:

Antoine Touzé pointed out to me the introductory book by Kraft and Procesi available at

http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf
In Chapter 4, p.32, by example, is a classical invariant results for matrices alluded to in Wenesday's talk:

For a permutation in the n-th symmetric group which is a cycle (i_1,...,i_k), associate the trace function defined to take a n-tuple of matrices (M_1,...M_n) to the trace of the matrix product M_i_1...M_i_k . For a permutation written as a product of disjoint cycles, take the product of the trace functions for each cycle in the decoñposition. We thus get a n-linear function for each permutation. These form a linear basis of the invariant n-linear functions on matrices under the (simultaneous) conjugation action of invertible matrices.