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'' $  M $
of a representation  $  \rho $
of a Lie algebra  $  \mathfrak t $
in a finite-dimensional vector space  $  V $''

The dimension  $  n _ {M} $
of the weight subspace  $  V _ {M} \subset  V $
corresponding to the weight  $  M $(
see [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]).

Let  $  \mathfrak t $
be a [[Cartan subalgebra|Cartan subalgebra]] of a semi-simple Lie algebra  $  \mathfrak g $
over an algebraically closed field of characteristic zero, and let  $  \rho $
be the restriction to  $  \mathfrak t $
of a finite-dimensional representation  $  \sigma $
of the algebra  $  \mathfrak g $. 
In this case the space  $  V $
is the direct sum of the weight subspaces of  $  \mathfrak t $
corresponding to the different weights. These weights and their multiplicities are often called the weights and the multiplicities of the representation  $  \sigma $
of the algebra  $  \mathfrak g $.

Suppose that  $  \sigma $
is an irreducible representation and let  $  \Lambda $
be its highest weight (see [[Cartan theorem|Cartan theorem]] on the highest weight vector). Then  $  n _  \Lambda  = 1 $. 
Various devices are available for computing the multiplicities of weights other than the highest weight. Two of these are classical results in representation theory: Freudenthal's formula and Kostant's formula.

1) Freudenthal's formula (see , [[#References|[1]]]). Let  $  (  , ) $
be the natural scalar product on the space  $  \mathfrak t  ^ {*} $
adjoint to  $  \mathfrak t $, 
induced by the [[Killing form|Killing form]] on  $  \mathfrak t $; 
let  $  R $
be the [[Root system|root system]] of the algebra  $  \mathfrak g $
relative to  $  \mathfrak t $
and let  $  > $
be a partial order relation on  $  \mathfrak t  ^ {*} $
determined by some fixed system of simple roots  $  \alpha _ {1} \dots \alpha _ {r} $
in  $  R $. 
Then

$$ 
(( \Lambda + \delta , \Lambda + \delta ) -
( M + \delta , M + \delta ) )
n _ {M\ } =
$$

$$ 
= \ 
2 \sum _ {\begin{array}{c}
\alpha \in R \\
 \alpha > 0 
\end{array}
 }  \sum _ {k = 1 } ^  \infty   n _ {M + k \alpha }  ( M + k \alpha , \alpha ),
$$

where  $  \delta = \sum _ {\alpha \in R, \alpha > 0 }  \alpha /2 $
and by definition  $  n _ {N} = 0 $
if  $  N $
is not a weight of the representation  $  \sigma $. 
For any weight  $  M \neq \Lambda $, 
the coefficient of  $  n _ {M} $
on the left of the formula is different from zero. This formula is essentially a recurrence formula: it enables one to express  $  n _ {M} $
in terms of  $  n _ {N} $
for  $  N > M $. 
Since it is known that  $  n _  \Lambda  = 1 $, 
Freudenthal's formula yields an effective method for the computation of the multiplicities  $  n _ {M }  $.

2) Kostant's formula (see [[#References|[5]]], [[#References|[1]]]). Let  $  \Gamma = \{ {M \in \mathfrak t  ^ {*} } : {2 ( M , \alpha _ {i} )/( \alpha _ {i} , \alpha _ {i} ) \in \mathbf Z  \textrm{ for  all  }   i = 1 \dots r } \} $. 
This set  $  \Gamma $
is a multiplicative subgroup in  $  \mathfrak t  ^ {*} $
which is invariant under the [[Weyl group|Weyl group]]  $  W $, 
which acts on  $  \mathfrak t  ^ {*} $
in a natural way. The element  $  \delta $— 
and indeed all weights of the representation  $  \sigma $— 
are members of  $  \Gamma $. 
Suppose that for each  $  M \in \Gamma $
the number  $  P ( M ) $
is the number of solutions  $  \{ {k _  \alpha  } : {\alpha \in R,  \alpha > 0 } \} $
of the equation

$$ 
M  = \ 
\sum _ {\begin{array}{c}
\alpha \in R \\
 \alpha > 0 
\end{array}
 }
k _  \alpha  \alpha ,
$$

where  $  k _  \alpha  \in \mathbf Z $, 
$  k _  \alpha  > 0 $
for all  $  \alpha $. 
The function  $  P ( M ) $
on  $  \Gamma $
is known as the partition function. Then

$$ 
n _ {M}  = \ 
\sum _ {S \in W }
(  \mathop{\rm det}  S) P ( S ( \Lambda + \delta ) -
( M + \delta )).
$$

Practical application of the above formulas involves cumbersome computations. For semi-simple algebras of rank 2, there are more convenient geometrical rules for evaluating the multiplicity of a weight (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.P. Zhelobenko,   "Lectures on the theory of Lie groups" , Dubna  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  H. Freudenthal,   "Zur Berechnung der Charaktere der halbeinfacher Liescher Gruppen I"  ''Indag. Math.'' , '''16'''  (1954)  pp. 369–376</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  H. Freudenthal,   "Zur Berechnung der Charaktere der halbeinfacher Liescher Gruppen II"  ''Indag. Math.'' , '''16'''  (1954)  pp. 487–491</TD></TR><TR><TD valign="top">[4c]</TD> <TD valign="top">  H. Freudenthal,   "Zur Berechnung der Charaktere der halbeinfacher Liescher Gruppen III"  ''Indag. Math.'' , '''18'''  (1956)  pp. 511–514</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B. Kostant,   "A formula for the multiplicity of a weight"  ''Trans. Amer. Math. Soc.'' , '''93'''  (1959)  pp. 53–73</TD></TR></table>

====Comments====
There is a faster algorithm for computing the full set of weights and multiplicities, due to M. Demazure [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Freudenthal,   H. de Vries,   "Linear Lie groups" , Acad. Press  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Demazure,   "Une nouvelle formule des charactères"  ''Bull. Sci. Math. (2)'' , '''98'''  (1974)  pp. 163–172</TD></TR></table>