weylAlcove -- the dominant integral weights of level less than or equal to l

Let $\mathbf{g}$ be a Lie algebra, and let $l$ be a nonnegative integer. Choose a Cartan subalgebra $\mathbf{h}$ and a base $\Delta= \{ \alpha_1,\ldots,\alpha_n\}$ of simple roots of $\mathbf{g}$. These choices determine a highest root $\theta$. (See highestRoot). Let $\mathbf{h}_{\mathbf{R}}^*$ be the real span of $\Delta$, and let $(,)$ denote the Killing form, normalized so that $(\theta,\theta)=2$. The fundamental Weyl chamber is $C^{+} = \{ \lambda \in \mathbf{h}_{\mathbf{R}}^* : (\lambda,\alpha_i) \ge 0, i=1,\ldots,n \}$. The fundamental Weyl alcove is the subset of the fundamental Weyl chamber such that $(\lambda,\theta) \leq l$. This function computes the set of integral weights in the fundamental Weyl alcove.

In the example below, we see that the Weyl alcove of $sl_3$ at level 3 contains 10 integral weights.