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Let $V$ be a vector space over some field. Is the natural representation $V$ of the group $GL(V)$ irreducible? Is it absolutely irreducible? Is the span of $GL(V)$ inside $End(V)$ all of $End(V)$?

I think the answer to the first two questions are yes, for the following reasons: the action of $GL(V)$ on $V-\{0\}$ is transitive; the center of $GL(V)$ is the group of all scalar multiples of $id_V$. Am I right?

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  • $\begingroup$ EDIT : For the first question, you are right. For the last question, I believe it to be true using elementary matrices and Gauss reduction algorithm. The center of $GL(V)$ is indeed the scalar matrices. $\endgroup$
    – Thibaut Dumont
    Apr 23, 2015 at 14:23

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