# Kostant partition function

(Redirected from Kostant's partition function)

In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system $\Delta$ is the number of ways one can represent a vector (weight) as an integral non-negative sum of the positive roots $\Delta^+\subset\Delta$. Kostant used it to rewrite the Weyl character formula for the multiplicity of a weight of an irreducible representation of a semisimple Lie algebra.

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

## Relation to the Weyl character formula

The values of Kostant's partition function are given by the coefficients of the power series expansion of

$\frac{1}{\prod_{\alpha>0}(1-e^{-\alpha})}$

where the product is over all positive roots. Using Weyl's denominator formula

${\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})},$

shows that the Weyl character formula

$\operatorname{ch}(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over \sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})}$

can also be written as

$\operatorname{ch}(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}.$

This allows the multiplicities of finite-dimensional irreducible representations in Weyl's character formula to be written as a finite sum involving values of the Kostant partition function, as these are the coefficients of the power series expansion of the denominator of the right hand side.