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| In [[representation theory]], a branch of mathematics, the '''Kostant partition function''', introduced by {{harvs|txt|authorlink=Bertram Kostant|first=Bertram |last=Kostant|year1=1958|year2=1959}}, of a [[root system]] <math>\Delta</math> is the number of ways one can represent a vector ([[Weight (representation theory)|weight]]) as an integral non-negative sum of the [[positive root]]s <math>\Delta^+\subset\Delta</math>. Kostant used it to rewrite the [[Weyl character formula]] for the [[multiplicity (mathematics)|multiplicity]] of a weight of an [[irreducible representation]] of a [[semisimple Lie algebra]].
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| The Kostant partition function can also be defined for [[Kac–Moody algebra]]s and has similar properties.
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| ==Relation to the Weyl character formula== | |
| The values of Kostant's partition function are given by the coefficients of the power series expansion of
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| :<math>\frac{1}{\prod_{\alpha>0}(1-e^{-\alpha})}</math> | |
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| where the product is over all positive roots. Using Weyl's denominator formula
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| :<math>{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})},</math>
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| shows that the [[Weyl character formula]]
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| :<math>\operatorname{ch}(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over \sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho})}</math>
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| can also be written as
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| :<math>\operatorname{ch}(V)={\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}) \over e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})}.</math>
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| This allows the multiplicities of finite-dimensional irreducible representations in Weyl's character formula to be written as a finite sum involving values of the Kostant partition function, as these are the coefficients of the power series expansion of the denominator of the right hand side.
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| ==References==
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| {{reflist}}
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| * Humphreys, J.E. Introduction to Lie algebras and representation theory, Springer, 1972.
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| *{{Citation | doi=10.1073/pnas.44.6.588 | last1=Kostant | first1=Bertram | title=A formula for the multiplicity of a weight | jstor=89667 | mr=0099387 | year=1958 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | issn=0027-8424 | volume=44 | pages=588–589 | issue=6 | publisher=National Academy of Sciences}}
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| *{{Citation | last1=Kostant | first1=Bertram | title=A formula for the multiplicity of a weight | jstor=1993422 | mr=0109192 | year=1959 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=93 | pages=53–73 | issue=1 | publisher=American Mathematical Society}}
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| [[Category:Representation theory]]
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| [[Category:Representation theory of Lie algebras]]
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| [[Category:Types of functions]]
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