Kostant 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

${\begin{aligned}{\frac {1}{\prod _{\alpha >0}(1-e^{-\alpha })}}&{}=\prod _{\alpha >0}(1+e^{-\alpha }+e^{-2\alpha }+e^{-3\alpha }+\cdots )\\&{}=\sum _{\lambda }p(\mu )e^{-\mu }\end{aligned}}$

where the product is over all positive roots, the sum is over elements $\mu$ of the root lattice, and $p(\mu )$ is the Kostant partition function. 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

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

can also be written as

$\operatorname {ch} (V)=(\sum _{w\in W}(-1)^{\ell (w)}e^{w(\lambda +\rho )-\rho })(\sum _{\mu }p(\mu )e^{-\mu }).$

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.

Sources

• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer
• Humphreys, J.E. Introduction to Lie algebras and representation theory, Springer, 1972.
• Kostant, Bertram (1958), "A formula for the multiplicity of a weight", Proceedings of the National Academy of Sciences of the United States of America, 44 (6), National Academy of Sciences: 588–589, doi:10.1073/pnas.44.6.588, ISSN 0027-8424, JSTOR 89667, MR 0099387
• Kostant, Bertram (1959), "A formula for the multiplicity of a weight", Transactions of the American Mathematical Society, 93 (1), American Mathematical Society: 53–73, doi:10.2307/1993422, ISSN 0002-9947, JSTOR 1993422, MR 0109192