# 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