Distance from a point to a line

In representation theory, a branch of mathematics, the Kostant partition function, introduced by Template:Harvs, of a root system $\Delta$ is the number of ways one can represent a vector (weight) as an integral non-negative sum of the positive roots $\Delta ^{+}\subset \Delta$. Kostant used it to rewrite the Weyl character 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.

Relation to the Weyl character formula

The values of Kostant's partition function are given by the coefficients of the power series expansion of

${\frac {1}{\prod _{\alpha >0}(1-e^{-\alpha })}}$

where the product is over all positive roots. Using Weyl's denominator formula

${\sum _{w\in W}(-1)^{\ell (w)}w(e^{\rho })=e^{\rho }\prod _{\alpha >0}(1-e^{-\alpha })},$

shows that the Weyl character formula

$\operatorname {ch} (V)={\sum _{w\in W}(-1)^{\ell (w)}w(e^{\lambda +\rho }) \over \sum _{w\in W}(-1)^{\ell (w)}w(e^{\rho })}$

can also be written as

$\operatorname {ch} (V)={\sum _{w\in W}(-1)^{\ell (w)}w(e^{\lambda +\rho }) \over e^{\rho }\prod _{\alpha >0}(1-e^{-\alpha })}.$

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.

References

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• Humphreys, J.E. Introduction to Lie algebras and representation theory, Springer, 1972.
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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