In the simplest terms, the Kostant partition function for an irreducible finite root system is a function which takes \(r\) natural numbers \(\mathbb{Z}_{\geq 0}\), where \(r\) is the rank/dimension of the root system, and returns a natural number.
A finite root system \(\Phi\) of rank \(r\) is a finite set of vectors in \(\mathbb{R}^r\) which have certain nice symmetry properties under reflection. A (non-canonical) choice of hyperplane divides the roots into positive and negative roots \(\Phi_+\) and \(\Phi_-\), and further selects a set of \(r\) simple roots \(\Phi_s\) which provide a basis for the space.
The positive root space is the positive integer span of the positive roots, and turns out to also be the positive integer span of the simple roots. Furthermore, each element of the positive root space has a unique expression in terms of the simple roots, but not necessarily in terms of the positive roots. The number of distinct ways to write the element of the positive root space in terms of positive roots is essentially the Kostant partition function.
Explicitly, if \(\{\alpha_i\}_{1 \leq i \leq r}\) are the simple roots, then each element of the positive root space can be written uniquely as \(\alpha = \sum_{i = 1}^r n_i \alpha_i\). Then the Kostant partition function \(P: \mathbb{N}^r \rightarrow \mathbb{N}\) is defined as the number of ways \(\alpha\) can be written as a non-negative integer combination of positive roots \(\sum_{i = 1}^{|\Phi_+|} \alpha_i\).
The simplest non-trivial case is given by the rank 2 root system \(A_2\), which has positive roots \(\alpha_1, \alpha_2, \alpha_3\), with simple roots \(\alpha_1, \alpha_2\) and with the algebraic relation \(\alpha_3 = \alpha_1 + \alpha_2\). In this case, it is not hard to see \(P(n_1, n_2) = 1 + \textrm{min}(n_1, n_2)\).
The other irreducible finite root systems of rank 2 are \(B_2\) and \(G_2\). The Kostant partition functions are known but are more involved, found for example in "Partition Function for Certain Simple Lie Algebras" (Tarski 1963).
If a weight \(\lambda\) is a weight of the representation \(V_\mu\) with highest weight \(\mu\), then the multiplicity of \(\lambda\) is given by the formula
\(\mathrm{mult}(\lambda) = \sum_{w\in W} \mathrm{det}(w) P(w \cdot (\mu + \delta) - (\lambda + \delta)),\)
where \(W\) is the Weyl group of the root system \(\Phi\) and \(\delta\) is the Weyl vector, which is half the sum of the positive roots.