Kostant partition function

In representation theory, a branch of mathematics, the Kostant partition function, introduced by , of a root system formula_1 is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots formula_2. 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.
An example.
Consider the A2 root systems, with positive roots formula_3, formula_4, and formula_5. If an element formula_6 can be expressed as a non-negative integer linear combination of formula_3, formula_4, and formula_9, then since formula_10, it can also be expressed as a non-negative integer linear combination of formula_3 and formula_4:
with formula_14 and formula_15 being non-negative integers. This expression gives "one" way to write formula_6 as a non-negative integer combination of positive roots; other expressions can be obtained by replacing formula_17 with formula_9 some number of times. We can do the replacement formula_19 times, where formula_20. Thus, if the Kostant partition function is denoted by formula_21, we obtain the formula
This result is shown graphically in the image at right. If an element formula_6 is not of the form formula_13, then formula_25.
Relation to the Weyl character formula.
Inverting the Weyl denominator.
For each root formula_26 and each formula_27, 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:
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 formula_31 can occur in the product.
Rewriting the character formula.
This argument shows that we can convert the Weyl character formula for the irreducible representation with highest weight formula_32:
from a quotient to a product:
The multiplicity formula.
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 formula_6 in the irreducible representation with highest weight formula_32:
This result is the Kostant multiplicity formula. 
The dominant term in this formula is the term formula_38; the contribution of this term is formula_39, which is just the multiplicity of formula_6 in the Verma module with highest weight formula_32. If formula_32 is sufficiently far inside the fundamental Weyl chamber and formula_6 is sufficiently close to formula_32, it may happen that all other terms in the formula are zero. Specifically, unless formula_45 is higher than formula_46, the value of the Kostant partition function on formula_47 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.