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

For each root $\alpha$ and each $H\in {\mathfrak {h}}$, we have can formally apply the formula for the sum of a geometric series to obtain

${\frac {1}{1-e^{-\alpha (H)}}}=1+e^{\alpha (H)}+e^{2\alpha (H)}+\cdots$

where do not worry about convergence—that is, the equality is understood at the level of formal power series. Using Weyl's denominator formula

${\sum _{w\in W}(-1)^{\ell (w)}w(e^{\rho (H)})=e^{\rho (H)}\prod _{\alpha >0}(1-e^{-\alpha (H)})},$

we obtain a formal expression for the reciprocal of the Weyl denominator:

${\begin{aligned}\sum _{w\in W}(-1)^{\ell (w)}w(e^{\rho (H)})&{}=e^{\rho (H)}\prod _{\alpha >0}(1+e^{-\alpha (H)}+e^{-2\alpha (H)}+e^{-3\alpha (H)}+\cdots )\\&{}=e^{\rho (H)}\sum _{\mu }p(\mu )e^{-\mu (H)}\end{aligned}}$

Here, the first equality is by taking a product over the positive roots of the geometric series formula and the second equality is by counting all the ways a given exponential $e^{\mu (H)}$ can occur in the product.

This argument 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. The result is a formula for the multiplicities of the weights of an irreducible finite dimensional representation, known as the Kostant multiplicity formula.

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