Kostant partition function

In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system ${\displaystyle \Delta }$ is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots ${\displaystyle \Delta ^{+}\subset \Delta }$. Kostant used it to rewrite the Weyl character formula as a formula (the Kostant multiplicity formula) for the multiplicity of a weight of an irreducible representation of a semisimple Lie algebra. An alternative formula, that is more computationally efficient in some cases, is Freudenthal's formula.

The Kostant partition function can also be defined for Kac–Moody algebras and has similar properties.