Root lattice ($Q$)

The root lattice is the integer span of the roots, equivalently of the simple roots.

The Kostant partition function $P(\nu )$ counts the ways to express a weight $\nu$ as a nonnegative integer combination of positive roots. By the Poincaré-Birkhoff-Witt theorem, $P(\lambda -\mu )$ is the multiplicity of the weight $\mu$ in the Verma module $M(\lambda )$.

The Kostant partition function counts nonnegative integer combinations of the positive roots of a root system, without ordering the summands. Multiply the formal geometric series ${\sum }_{k\ge 0}{e}^{-k\alpha }$ for every positive root. The coefficient of $e^{-\nu }$ counts precisely these combinations. Every coefficient is finite because a linear functional strictly positive on the positive roots bounds every participating integer coefficient. The sign of each exponent in the cancelling product must agree with that of this series.

For weights $\mu ,\lambda$, one writes $\mu ⪯\lambda$ when $\lambda -\mu$ is a nonnegative integer combination of the simple roots.