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 nonexceptional type for which the first and last Dynkin node did not give an irrep with onedimensional weight spaces. Is there any conceptual explanation for this?
$\begingroup$
$\endgroup$
2

1$\begingroup$ 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. $\endgroup$– Mikko KorhonenMar 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? $\endgroup$– Fofi KonstantopoulouMar 9, 2020 at 23:05
Add a comment

1 Answer
$\begingroup$
$\endgroup$
2
The LiE software does these calculations, and is available on line:

$\begingroup$ 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

2$\begingroup$ 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