Kostant partition function
Introduction 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 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. 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 ===The Kostant partition function for the A2 root systemValues of the Kostant partition function for the root system . The root system is given the Euclidean coordinates .Consider the A2 root system, with positive roots , , and . If an element  can be expressed as a non-negative integer linear combination of , , and , then since , it can also be expressed as a non-negative integer linear combination of the positive simple roots  and :: with  and  being non-negative integers. This expression gives ''one'' way to write  as a non-negative integer combination of positive roots; other expressions can be obtained by replacing  with  some number of times. We can do the replacement  times, where . Thus, if the Kostant partition function is denoted by , we obtain the formula:.This result is shown graphically in the image at right. If an element  is not of the form , then .=== B2 ===The partition function for the other rank 2 root systems are more complicated but are known explicitly.For B2, the positive simple roots are , and the positive roots are the simple roots together with  and . The partition function can be viewed as a function of two non-negative integers  and , which represent the element . Then the partition function  can be defined piecewise with the help of two auxiliary functions.If , then . If , then . If , then . The auxiliary functions are defined for  and are given by  and  for  even,  for  odd.=== G2 ===For G2, the positive roots are  and , with  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.
Relation to the Weyl character formula ===Inverting the Weyl denominator===For each root  and each , 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  can occur in the product. The function  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 for the irreducible representation with highest weight ::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  in the irreducible representation with highest weight ::.This result is the '''Kostant multiplicity formula'''. The dominant term in this formula is the term ; the contribution of this term is , which is just the multiplicity of  in the Verma module with highest weight . If  is sufficiently far inside the fundamental Weyl chamber and  is sufficiently close to , it may happen that all other terms in the formula are zero. Specifically, unless  is higher than , the value of the Kostant partition function on  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. **