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All the lie algebra considered are over $\mathbf{C}$. I know that for the lie algebra $\mathfrak{sl}_{n+1}$ the fundamental representations $L(\omega_k), k \in \{1,\cdots,n\}$ are the $\Lambda^k V$ where $V = \mathbf{C}^{n+1}$ is the natural representation.

By searching on google a little bit it seems that for other classical Lie algebras there aren't such nice results.

Still i'm wondering what can one say. For example is the natural representation always a fundamental representation ? The adjoint representation ?

Also is the problem easier for Lie algebras of rank $2$ ? The reason I ask is that in a past exam of my Lie algebra class they asked to find the fundamental representations of G2. They first make us show that the adjoint representation is a fundamental representation and then they ask us to show that the other representation is of dimension $7$. I have no idea how to do that last part.

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The natural representation $L(\omega_1)$ is fundamental for type $A_l,B_l,C_l,D_l$. Most of the other fundamental representations can be obtained by considering exterior powers of the natural representation. In general however, the situation is more complicated than in the case of type $A_l$, where the exterior powers $\Lambda^iV$ exhaust all fundamental $L$-modules, for $i=1,\ldots ,l$. Namely, for type $B_l$ the exterior powers $\Lambda^iV$ are still fundamental $L$-modules for $i=1,\ldots ,l-1$, but there is one remaining fundamental module, which is not of this form, i.e., $L(\omega_l)$, the spin module of dimension $2^l$. For type $C_l$ the exterior powers are not irreducible in general, so one has to take the kernels of the contractions. For type $D_l$, the exterior powers are fundamental $L$-modules for $i=1,\ldots ,l-2$, and there are two additional fundamental modules $L(\omega_l)$ and $L(\omega_{l-1})$, both of dimension $2^{l-1}$. For $G_2$ we have fundamental modules of dimension $7$ and $14$. For $E_8$ the lowest-dimensional one is already the adjoint representation of dimension $248$. There is a large literature on this for exceptional simple Lie algebras.

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  • $\begingroup$ This is a great answer thank you very much. A few precision though : I know that the adjoint representation of G2 is fundamental (it's $L(\omega_1)$ if my computations are correct) but how dooes one see that the other one is of dimension 7 ? Also (and this might be related since the character gives you the dimension) is there a way to compute the character of the fundamental representations without being able to construct them explicitely ? $\endgroup$ – A.champolion Oct 20 '15 at 6:33
  • $\begingroup$ It comes from viewing $G_2$ as the automorphism group of the octonions, see here, so that $G_2\subseteq O(\mathbb{R}^7)$. $\endgroup$ – Dietrich Burde Oct 20 '15 at 8:42
  • $\begingroup$ Ok thanks a lot. I'm pushing my luck here but do you by any chance know an answer to my question about characters : i.e. is there a way to compute the character of the fundamental representations without knowing them explicitely ? $\endgroup$ – A.champolion Oct 20 '15 at 10:31
  • $\begingroup$ Ah I just learned about Weyl character and dimension formula which are what I was looking for, I think. A lot of things (although without any proofs) are well explained I find in the book lectures on representation theory by Huang (if anybody ever reads this). Anyway thanks a lot again for your answer Dietrich Burde. $\endgroup$ – A.champolion Oct 20 '15 at 16:16

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