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Will the fundamental representation $\pi_n$ of type $C_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be calculated? If so this would make $C_n$ the only non-exceptional type for which the first and last Dynkin node did not give an irrep with one-dimensional weight spaces. Is there any conceptual explanation for this?

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    $\begingroup$ I think the answer is yes, some weight space will have dimension $> 1$, unless you are taking about groups of type $C_n$ in characteristic two. See this question. Frank Lübeck's website might also be useful: link. $\endgroup$ Mar 4, 2020 at 13:30
  • $\begingroup$ @Mikko: In the book you referenced in the linked question, it seems the author claims that C_n has $1$-dim weight spaces, for all $n$. Is this an error, or am I reading the notation incorrectly? $\endgroup$ Mar 9, 2020 at 23:05

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The LiE software does these calculations, and is available on line:

http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html

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  • $\begingroup$ I don't see an explicit weight space function. How can one deduce this from the given functions? $\endgroup$ Mar 7, 2020 at 17:44
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    $\begingroup$ Choose Full character from the pulldown menu for multiplicities of all weights, or Character multiplicities for just the multiplicities of the dominant weights. $\endgroup$ Mar 8, 2020 at 19:42

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