Multiplicity of a weight

$ 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