The Kostant partition function

Let ${\mathrm{\Phi }}^{+}$ be a positive system for the finite root system of Finite Weyl root system, lattice and chamber conventions with simple roots ${\alpha }_1,\dots ,{\alpha }_r$ and positive cone $Q_{+}$. For $\beta \in {\mathfrak{h}}^{\ast }$ define $P(\beta )$ to be the number of families $(n_{\alpha }{)}_{\alpha \in {\mathrm{\Phi }}^{+}}\in {\mathbb{Z}}_{\ge 0}^{{\mathrm{\Phi }}^{+}}$ with ${\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{n}_{\alpha }\alpha =\beta \mathrm{.}$ This number is finite: writing an element of $Q$ in the simple-root basis as $\beta ={\sum }_{i=1}^r{m}_i{\alpha }_i$ by Simple roots form a signed integral basis, the weight $\beta$ can be represented only if every $m_i\ge 0$, and then each $n_{\alpha }$ is bounded by the height $\mathrm{ht}(\beta )={\sum }_i{m}_i$ extending the root height of Height and highest root to $Q$ by the same coordinate sum, because every positive root has height at least $1$; so only finitely many families occur, and $P(\beta )=0$ for $\beta \mathrm{\notin }{Q}_{+}$ while $P(0)=1$ (the all-zero family, empty when ${\mathrm{\Phi }}^{+}=\mathrm{\varnothing }$).

Equivalently, $P(\beta )$ is the coefficient of $e^{-\beta }$ in the finite product of geometric series ${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}{\sum }_{k\ge 0}{e}^{-k\alpha }={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ of Geometric series are invertible in the completed character ring: a family $(n_{\alpha })$ with ${\sum }_{\alpha }{n}_{\alpha }\alpha =\beta$ contributes one monomial $e^{-\beta }$, and a family representing $\beta$ has $n_{\alpha }\le \mathrm{ht}(\beta )$ and hence finite support, so ${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}={\sum }_{\beta \in Q_{+}}P(\beta )e^{-\beta }$ in the completed character ring $\mathcal{R}$ of The completed formal character ring.