Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\mathfrak{sp_n}$ generates all the other fundamental representations? By generates I mean that all other fundamental representations are contained in some tensor power $V_1^{\otimes k}$?
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5$\begingroup$ Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial representation, then $\bigwedge^k V_1 = \bigoplus_{0\leq\ell\leq k,\;\ell\equiv k\pmod{2}} V_\ell$ (no multiplicities) for $0\leq k\leq n$. This is certainly well-known, but sadly I don't have a reference. $\endgroup$– Gro-TsenCommented Mar 27 at 21:21
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5$\begingroup$ Ah, here's a reference: Fulton & Harris, Representation Theory: A First Course (Springer GTM 129), theorem 17.5. $\endgroup$– Gro-TsenCommented Mar 27 at 21:27
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$\begingroup$ @Gro-Tsen: Thanks a lot for the answer and reference. If you put it as an answer then I can accept it. $\endgroup$– Zoltan FleishmanCommented Mar 27 at 21:40
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$\begingroup$ (Of course, this is only talking about finite-dimensional repns...) $\endgroup$– paul garrettCommented Mar 27 at 22:00
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2$\begingroup$ All irreducible algebraic representations of the symplectic group, full stop, are contained in some tensor power of $V_1$. $\endgroup$– Dan PetersenCommented Mar 28 at 9:17
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(Copied from my own comments.) Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial representation, then for $0\leq k\leq n$ we have $$\bigwedge^k V_1 = \bigoplus_{0\leq\ell\leq k,\;\ell\equiv k\pmod{2}} V_\ell$$ (no multiplicities). A possible reference is: : Fulton & Harris, Representation Theory: A First Course (Springer GTM 129), theorem 17.5.
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$\begingroup$ For completeness of MathOverflow, compare with: this question which is about the symmetric powers of the same standard (=first fundamental) representation $V_1$. $\endgroup$– Gro-TsenCommented Aug 27 at 8:08