  Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A
layer sum is introduced as the sum of formal exponentials of the distinct
weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that
the character of every finite-dimensional irreducible $\mathfrak{g}$-module
admits a decomposition in terms of layer sums, with only non-negative integer
coefficients. Ensuing results include a new approach to the computation of Weyl
characters and weight multiplicities, and a closed-form expression for the
number of distinct weights in a finite-dimensional irreducible
$\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin
labels, of degree equal to the rank of $\mathfrak{g}$.
