In representation theory, a branch of mathematics, the Kostant partition function, introduced by , of a root system is the number of ways one can represent a vector (weight) as a non-negative integer combination of the positive roots. 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.

Examples

A2

can be expressed as a non-negative integer linear combination, it can also be expressed as a non-negative integer linear combination of the positive simple roots:

being non-negative integers. This expression gives one, we obtain the formula

1. .

.

B2

The partition function for the other rank 2 root systems are more complicated but are known explicitly.

For B2 can be defined piecewise with the help of two auxiliary functions.

odd.

G2

For G2 denoting the long simple root.

The partition function is defined piecewise with the domain divided into five regions, with the help of two auxiliary functions.

Relation to the Weyl character formula

Inverting the Weyl denominator

, we can formally apply the formula for the sum of a geometric series to obtain

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

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

is zero if the argument is a rotation and one if the argument is a reflection.

Rewriting the character formula

This argument shows that we can convert the Weyl character formula:

from a quotient to a product:

The multiplicity formula

:

1. .

This result is the Kostant multiplicity formula.

in the Verma module will be zero. Thus, although the sum is nominally over the whole Weyl group, in most cases, the number of nonzero terms is smaller than the order of the Weyl group.

References

Sources

• Humphreys, J.E. Introduction to Lie algebras and representation theory, Springer, 1972.