The <formal character of a weight module> is $\operatorname{ch}V=\sum_\mu(\dim V_\mu)e^\mu$, where $e^\mu$ are formal basis symbols in the <group algebra> of the <weight lattice>, satisfying $e^\mu e^\nu=e^{\mu+\nu}$. For a finite-dimensional irreducible <highest-weight representation>, the <highest weight> has <weight multiplicity> one. Another standard result is that <weight multiplicities> are invariant under the <Weyl group>: $\dim V_{w\mu}=\dim V_\mu$.

A <minuscule representation> has all its weights in a single <Weyl group> orbit. Since the <highest weight> $\lambda$ is present, that orbit is $W\lambda$, and the invariance just stated shows every point of it is present with multiplicity one. Consequently the <formal character of a minuscule representation> is
$$
\boxed{\operatorname{ch}V(\lambda)=\sum_{\mu\in W\lambda}e^\mu,\qquad\dim V(\lambda)=|W\lambda|.}
$$
The sum is over distinct weights, rather than over all elements of $W$: summing $e^{w\lambda}$ over $w\in W$ would count each weight $|\operatorname{Stab}_W(\lambda)|$ times.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 102 / 5 / a / Solution

The <formal character of a weight module> for a <minuscule representation> of <highest weight> $\lambda$ is $\operatorname{ch}V(\lambda)=\sum_{\mu\in W\lambda}e^\mu$. The sum is over distinct points of the <Weyl group> orbit, so $\dim V(\lambda)=|W\lambda|$.

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