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Dec 3, 2011 at 15:38 comment added David Wehlau @Dev: You are welcome. Looking back at my answer I realize I misread question 2. Let i run over an index set of size k. Some things may be known for k=2 but I don't think anything is known for larger values of k except there is a complete answer for general k for $GL(2,F)$. See the paper [Vector invariants for the two dimensional Modular representation of a cyclic group of prime order, Campbell, Shank, Wehlau, Adv. in Math., 225(2) 1069-1094].
Jul 28, 2011 at 3:27 history answered David Wehlau CC BY-SA 3.0