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  • I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate.

    But $U(n)$ being a rank $n$ group shouldn't there be $n$ fundamental representations of it corresponding to the $n$ fundamental weights of it (dual to its $n$ simple roots)?

  • In the fundamental representation I think of the Cartan of ${\cal u}(n)$ to be spanned by the $n$ diagonal matrices of ${\cal u}(n)$ which have all $0$s except a $1$ then I guess the $n$ vectors $(0,..,1,..0)$ (the $1$ shifting through the $n$ positions) can be thought of as the $n$ weight-vectors of the representation?

    • And the same vectors with the $1$ replaced by $-1$ be thought of as the weight vectors of the anti-fundamental representation? (since conjugate transpose of any element of ${\cal u}(n)$ is negative of it?)

    Naively the above does seem to depend on whether I think of the Lie algebra of $U(n)$ to be $n\times n$ Hermitian or skew-Hermitian matrices depending on whether or not I have an "$i$" while taking the exponentiation from the Lie algebra.

    It would be helpful if someone can help disentangle this (possibly there is being a mix of what is an intrinsic property of the group and what is convention)

    • Is there an analogoue of the above construction for $U(n)$?
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    What is a fundamental representation (in the phrase "$n$ fundamental representations")?2012-08-09
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    @Qiachu Yuan I guess one definition of a "fundamental representation" is that these are those whose highest weights are fundamental weights. Now there is one fundamental weight corresponding to every simple co-root and there will be as many simple co-roots as the rank of the Lie algebra and hence my guess about that count.2012-08-09

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