# Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$

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?

• 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. Mar 4, 2020 at 13:30
• @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? Mar 9, 2020 at 23:05

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

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

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