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Aug 13, 2015 at 14:34 comment added darij grinberg By "simultaneous eigen-vectors" you probably mean "simultaneous eigenvectors for eigenvalue $\lambda_1 \lambda_2 \cdots \lambda_n$"? And the action of the torus on $F\left(V\right)$ should just follows from functoriality, via the embedding of the torus $T$ into $\operatorname{GL}\left(V\right)$ as diagonal matrices.
Aug 13, 2015 at 12:16 history answered F. C. CC BY-SA 3.0