Kostant partition function

In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system

is the number of ways one can represent a vector (weight) as a non-negative integer linear 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.

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

can be expressed as a non-negative integer linear combination of

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

This expression gives one way to write

as a non-negative integer combination of positive roots; other expressions can be obtained by replacing

Thus, if the Kostant partition function is denoted by

, we obtain the formula This result is shown graphically in the image at right.

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

[1][2] For B2, the positive simple roots are

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

The auxiliary functions are defined for

denoting the short simple root and

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.

, 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:[3] 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

This argument shows that we can convert the Weyl character formula for the irreducible representation with highest weight

: from a quotient to a product: Using the preceding rewriting of the character formula, it is relatively easy to write the character as a sum of exponentials.

The coefficients of these exponentials are the multiplicities of the corresponding weights.

We thus obtain a formula for the multiplicity of a given weight

in the irreducible representation with highest weight

:[4] This result is the Kostant multiplicity formula.

in the Verma module with highest weight

is sufficiently far inside the fundamental Weyl chamber and

, it may happen that all other terms in the formula are zero.

, the value of the Kostant partition function on

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.