Multiplicity of a weight

of a representation of a Lie algebra in a finite-dimensional vector space

The dimension of the weight subspace corresponding to the weight (see Weight of a representation of a Lie algebra).

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

Suppose that is an irreducible representation and let be its highest weight (see Cartan theorem on the highest weight vector). Then . 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 adjoint to , induced by the Killing form on ; let be the root system of the algebra relative to and let be a partial order relation on determined by some fixed system of simple roots in . Then

where and by definition if is not a weight of the representation . For any weight , the coefficient of on the left of the formula is different from zero. This formula is essentially a recurrence formula: it enables one to express in terms of for . Since it is known that , Freudenthal's formula yields an effective method for the computation of the multiplicities .

2) Kostant's formula (see [5], [1]). Let . This set is a multiplicative subgroup in which is invariant under the Weyl group , which acts on in a natural way. The element — and indeed all weights of the representation — are members of . Suppose that for each the number is the number of solutions of the equation

where , for all . The function on is known as the partition function. Then

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