In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant (1958, 1959), of a root system Δ {\displaystyle \Delta } is the number of ways one can represent a vector (weight) as a non-negative integer combination of the positive roots Δ + ⊂ Δ {\displaystyle \Delta ^{+}\subset \Delta } . 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
Consider the A2 root system, with positive roots α 1 {\displaystyle \alpha _{1}} , α 2 {\displaystyle \alpha _{2}} , and α 3 := α 1 + α 2 {\displaystyle \alpha _{3}:=\alpha _{1}+\alpha _{2}} . If an element μ {\displaystyle \mu } can be expressed as a non-negative integer linear combination of α 1 {\displaystyle \alpha _{1}} , α 2 {\displaystyle \alpha _{2}} , and α 3 {\displaystyle \alpha _{3}} , then since α 3 = α 1 + α 2 {\displaystyle \alpha _{3}=\alpha _{1}+\alpha _{2}} , it can also be expressed as a non-negative integer linear combination of the positive simple roots α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} :
μ = n 1 α 1 + n 2 α 2 {\displaystyle \mu =n_{1}\alpha _{1}+n_{2}\alpha _{2}}
with n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} being non-negative integers. This expression gives one way to write μ {\displaystyle \mu } as a non-negative integer combination of positive roots; other expressions can be obtained by replacing α 1 + α 2 {\displaystyle \alpha _{1}+\alpha _{2}} with α 3 {\displaystyle \alpha _{3}} some number of times. We can do the replacement k {\displaystyle k} times, where 0 ≤ k ≤ m i n ( n 1 , n 2 ) {\displaystyle 0\leq k\leq \mathrm {min} (n_{1},n_{2})} . Thus, if the Kostant partition function is denoted by p {\displaystyle p} , we obtain the formula
p ( n 1 α 1 + n 2 α 2 ) = 1 + m i n ( n 1 , n 2 ) {\displaystyle p(n_{1}\alpha _{1}+n_{2}\alpha _{2})=1+\mathrm {min} (n_{1},n_{2})} . This result is shown graphically in the image at right. If an element μ {\displaystyle \mu } is not of the form μ = n 1 α 1 + n 2 α 2 {\displaystyle \mu =n_{1}\alpha _{1}+n_{2}\alpha _{2}} , then p ( μ ) = 0 {\displaystyle p(\mu )=0} .
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