Dimension of weight spaces of Lie algebra representation
Consider a Lie algebra g. Let λ be a dominant integral weight and L(λ) be the unique irreducible representation of highest weight λ. (Since λ is dominant and integral, L(λ) is finite dimensional).
We know that L(λ) decomposes into a direct sum L(λ)=⨁μL(λ)μ where L(λ)μ is a weight space of weight μ.
Is there a way to compute dimL(λ)μ in Sage?
I know that Freudenthal formula can be used to find these dimensions by hand. But I want to verify if my calculations are correct. Thanks in advance!