Atomic Decompositions of Lie Characters and the Dominant Weight Poset

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $λ$, let $Θ_λ$ be the multiplicity-free sum of the weights of the irreducible module $V(λ)$. The $Θ_μ$ form a $\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(μ,λ)$, the atomic numbers