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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655101.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655102.png" /> of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655103.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655104.png" />''
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The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655105.png" /> of the weight subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655106.png" /> corresponding to the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655107.png" /> (see [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]).
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 +
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655108.png" /> be a [[Cartan subalgebra|Cartan subalgebra]] of a semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m0655109.png" /> over an algebraically closed field of characteristic zero, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551010.png" /> be the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551011.png" /> of a finite-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551012.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551013.png" />. In this case the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551014.png" /> is the direct sum of the weight subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551015.png" /> corresponding to the different weights. These weights and their multiplicities are often called the weights and the multiplicities of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551016.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551017.png" />.
+
'' $  M $
 +
of a representation  $  \rho $
 +
of a Lie algebra $  \mathfrak t $
 +
in a finite-dimensional vector space $  V $''
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551018.png" /> is an irreducible representation and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551019.png" /> be its highest weight (see [[Cartan theorem|Cartan theorem]] on the highest weight vector). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551020.png" />. 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.
+
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]]).
  
1) Freudenthal's formula (see , [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551021.png" /> be the natural scalar product on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551022.png" /> adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551023.png" />, induced by the [[Killing form|Killing form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551024.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551025.png" /> be the [[Root system|root system]] of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551026.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551027.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551028.png" /> be a partial order relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551029.png" /> determined by some fixed system of simple roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551031.png" />. Then
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551032.png" /></td> </tr></table>
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551033.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551034.png" /> and by definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551035.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551036.png" /> is not a weight of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551037.png" />. For any weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551038.png" />, the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551039.png" /> on the left of the formula is different from zero. This formula is essentially a recurrence formula: it enables one to express <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551040.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551042.png" />. Since it is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551043.png" />, Freudenthal's formula yields an effective method for the computation of the multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551044.png" />.
+
$$
 +
(( \Lambda + \delta , \Lambda + \delta ) -
 +
( M + \delta , M + \delta ) )
 +
n _ {M\ } =
 +
$$
  
2) Kostant's formula (see [[#References|[5]]], [[#References|[1]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551045.png" />. This set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551046.png" /> is a multiplicative subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551047.png" /> which is invariant under the [[Weyl group|Weyl group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551048.png" />, which acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551049.png" /> in a natural way. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551050.png" /> — and indeed all weights of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551051.png" /> — are members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551052.png" />. Suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551053.png" /> the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551054.png" /> is the number of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551055.png" /> of the equation
+
$$
 +
= \
 +
2 \sum _ {\begin{array}{c}
 +
\alpha \in R \\
 +
\alpha > 0  
 +
\end{array}
 +
}  \sum _ {k = 1 } ^  \infty  n _ {M + k \alpha }  ( M + k \alpha , \alpha ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551056.png" /></td> </tr></table>
+
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 }  $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551058.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551059.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551060.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551061.png" /> is known as the partition function. Then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065510/m06551062.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {\begin{array}{c}
 +
\alpha \in R \\
 +
\alpha > 0
 +
\end{array}
 +
}
 +
k _  \alpha  \alpha ,
 +
$$
 +
 
 +
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]]]).
 
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]]]).
Line 27: Line 118:
 
====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 08:02, 6 June 2020


$ 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).

Let $ \mathfrak t $ be a 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 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 , [1]). Let $ ( , ) $ be the natural scalar product on the space $ \mathfrak t ^ {*} $ adjoint to $ \mathfrak t $, induced by the Killing form on $ \mathfrak t $; let $ R $ be the 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 [5], [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 $ 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 [2]).

References

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

Comments

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

References

[a1] H. Freudenthal, H. de Vries, "Linear Lie groups" , Acad. Press (1969)
[a2] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4
[a3] M. Demazure, "Une nouvelle formule des charactères" Bull. Sci. Math. (2) , 98 (1974) pp. 163–172
How to Cite This Entry:
Multiplicity of a weight. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicity_of_a_weight&oldid=11685
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article