UDC 519.46

MATHEMATICS

A. M. PERELOMOV, V. S. POPOV

CASIMIR OPERATORS FOR CLASSICAL GROUPS

(Presented by Academician I. G. Petrovskii, 7 VII 1966)

Invariant operators constructed from the generators of a group (Casimir operators \((^1)\)) are important both for group theory itself and for applications. These operators were first considered in general form by I. M. Gelfand \((^2)\) and Racah \((^3)\). Although an extensive literature has been devoted to their properties (see, for example, \((^{2-7})\)), explicit formulas for the eigenvalues of Casimir operators of arbitrary order had not previously been obtained. Below we give a solution of this problem for all “classical” groups (in the terminology of H. Weyl).

Let us first consider the groups \(U(n)\) and \(SU(n)\). Their generators satisfy the commutation relations

\[ [A_j^i,\ A_l^k]=\delta_j^k A_l^i-\delta_l^i A_j^k . \tag{1} \]

The Casimir operator of order \(p\) has the form

\[ C_p=\sum_{i_1,\ldots,i_p} A_{i_2}^{i_1} A_{i_3}^{i_2}\ldots A_{i_1}^{i_p}. \tag{2} \]

An irreducible representation is specified by the Young diagram \(\{f_1,f_2,\ldots,f_n\}\), \(f_1\geq f_2\geq \cdots \geq f_n\geq 0\); here \(f_i\) is the number of boxes in the \(i\)-th row. Since the operator \(C_p\) in any irreducible representation reduces to a constant, to find its eigenvalue \(C_p(f_1,f_2,\ldots,f_n)\) we apply \(C_p\) to the highest vector of the representation, satisfying the relations

\[ A_i^i\psi_0=m\psi,\qquad A_j^i\psi=0\quad \text{for } i<j . \tag{3} \]

The numbers \(m_i\) are the components of the highest weight of the irreducible representation;

\[ m_i=f_i \text{ for the group } U(n),\qquad m_i=f_i-\frac{1}{n}(f_1+f_2+\cdots+f_n) \text{ for the group } SU(n). \]

Rewriting \(C_p\) in the form

\[ C_p=\sum_{i,j=1}^{n} (T_{p-1})_j^i A_i^j,\qquad \text{where }\ (T_{p-1})_j^i=\sum_{i_1,\ldots,i_{p-2}=1}^{n} A_{i_1}^{i} A_{i_2}^{i_1}\ldots A_j^{i_{p-2}}, \tag{4} \]

we note that the operator \((T_{p-1})_j^i\) has the same transformation properties as the generator \(A_j^i\). Hence

\[ C_p\psi_0=\sum_{i=1}^{n}(m_i+2r_i)(T_{p-1})_i^i\psi, \tag{5} \]

\[ (T_q)_i^i\psi=\sum_{j=1}^{n} a_{ij}(T_{q-1})_j^j\psi . \tag{6} \]

Here \(r_i=(n+1)/2-i\) (the half-sum of the positive roots of the unitary group),

\[ a_{ij}=(m_i+n-i)\delta_{ij}-\theta_{ij},\qquad \theta_{ij}= \begin{cases} 1, & \text{for } i<j,\\ 0, & \text{for } i\geq j . \end{cases} \tag{7} \]

Proceeding recursively, we arrive at the final result \((^8,^9)\)

\[ C_p(f_1,f_2,\ldots,f_n)=\sum_{i,j}(a^p)_{ij}=\operatorname{Sp}(a^pE), \tag{8} \]

where the matrix \(E\), consisting entirely of ones, has been introduced: \(E_{ij}=1\) for all \(i,j\). If the matrix \(a\) is brought to diagonal form, then we obtain the following expression for \(C_p\):

\[ C_p(f_1,f_2,\ldots,f_n)=\sum_{i=1}^{n}\lambda_i^p\prod_{j\ne i}\frac{\lambda_i-\lambda_j-1}{\lambda_i-\lambda_j}, \qquad \lambda_i=m_i+n-i . \tag{9} \]

Formula (9) represents \(C_p(f_1,f_2,\ldots,f_n)\) in a uniform way for all values of \(p\).* Its drawback, however, is that the individual terms in (9) are rational functions of the variables \(\lambda_i\), whereas \(C_p(f_1,f_2,\ldots,f_n)\) as a whole reduces to a polynomial. For concrete computations it is convenient to expand \(C_p\) in a series in the power sums

\[ S_k=\sum_{i=1}^{n}(\lambda_i^k-\rho_i^k), \]

\(\rho_i=n-i\) (for the identity representation all \(S_k\) vanish). To obtain such an expansion, observe that (9) can be rewritten in the form of a contour integral

\[ C_p(f_1,f_2,\ldots,f_n) = -\frac{1}{2\pi i}\oint_{(+)} \lambda^p \prod_{i=1}^{n}\left(1-\frac{1}{\lambda-\lambda_i}\right)\,d\lambda, \tag{10} \]

from which the form of the generating function for the Casimir operators follows immediately:

\[ G(z)=\sum_{p=0}^{\infty} C_p z^p = n e^{-\varphi(z)}+\frac{1-e^{-\varphi(z)}}{z}, \tag{11} \]

where

\[ \varphi(z)=\sum_{k=2}^{\infty}a_k z^k, \qquad a_k=\sum_{l=1}^{k-1}\frac{(k-1)!}{l!(k-l)!}S_l . \tag{12} \]

Introduce the quantities \(B_p\), defined by the expansion

\[ e^{-\varphi(z)}=-\sum_{p=0}^{\infty}B_{p-1}z^p, \tag{13} \]

\[ B_{-1}=-1,\quad B_0=0,\quad B_1=S_1,\quad B_2=S_2+S_1,\quad B_3=S_3+\frac{3}{2}S_2-\frac{1}{2}S_1^2+S_1, \]

\[ B_4=S_4+2S_3-S_2S_1+2S_2-S_1^2+S_1,\ldots \]

and so on. Then the final formula for the eigenvalues of the Casimir operators is written in the form

\[ C_p(f_1,f_2,\ldots,f_n)=B_p-nB_{p-1}. \tag{14} \]

The computations are carried out analogously for the remaining classical groups. The Casimir operator is still defined by formula (2); acting with it on the highest vector of an irreducible representation specified by the highest weight \(\mathbf{m}=(m_1,m_2,\ldots,m_n)\), \(m_1\ge m_2\ge\cdots\ge m_n\), for the eigenvalue \(C_p(\mathbf{m})\) we obtain** formula (8), into which one must substitute the appropriate matrix \(a_{ij}\) for each of the groups:

\[ a_{ij}=(l_i+\alpha)\delta_{ij}-\theta_{ij}+\beta\frac{1+\varepsilon_i}{2}\delta_{i,-j}, \tag{15} \]

* The importance of representing \(C_p\) in the form (9) was pointed out to us by I. M. Gelfand.

** A more detailed presentation of the computations can be found in \((^9)\).

where \(l_i=m_i+r_i\) \((i>0)\)*, \(l_{-i}=-l_i\); the quantities \(\alpha,\ \beta,\ r_i\) are given in Table 1, \(\theta_{ij}\) is an upper triangular matrix all of whose elements standing above the main diagonal are equal to one (with \(\theta_{ii}=0\) for any \(i\)), and the quantities \(\varepsilon_i\) are equal to

Table 1

\[ \varepsilon_i= \begin{cases} 1 & \text{for } i>0,\\ 0 & \text{for } i=0,\\ -1 & \text{for } i<0. \end{cases} \tag{16} \]

Formulas (8), (15) reduce the computation of the Casimir operators to the elementary problem of raising the known matrix \(a_{ij}\) to the \(p\)-th power. For small values of \(p\) the authors found \((^{8,9})\) more convenient expressions in terms of the power sums \(S_k\).

Acting by the same method as in the case of the unitary group, one can bring \(C_p\) to a form analogous to (9). These formulas, as well as expressions for the generating function of the Casimir operators for the orthogonal and symplectic groups, will be considered in the authors’ next work.

As is known, in a group of rank \(n\) there are exactly \(n\) independent Casimir operators. It can be shown \((^{7,9})\) that these are \(C_1,C_2,\ldots,C_n\) for the group \(U(n)\) and \(C_2,C_4,\ldots,C_{2n}\) for the groups \(B_n,C_n\), and \(D_n\). For all classical groups except \(O(2n)\), the listed Casimir operators uniquely determine the irreducible representation. The peculiarity of the group \(O(2n)\) is that \((^{10})\) it has two inequivalent spinor representations \(\Delta_+\) and \(\Delta_-\), whose highest weights differ only in the sign of the last component: \(m_{\pm}=(1/2,\ldots,1/2,\pm 1/2)\). The eigenvalues of all operators \(C_{2k}\) on the representations \(\Delta_+\) and \(\Delta_-\) coincide with one another; therefore, in order to restore a one-to-one correspondence between the weight vector \(\mathbf m\) and the set of Casimir operators, one has to introduce a scalar operator \(\tilde C\), different from (2). Such an operator for the group \(O(2n)\) is the pseudoscalar

\[ \tilde C=\sum_{(i,j)} \varepsilon^{j_1j_2\ldots j_n}_{i_1i_2\ldots i_n} A^{i_1}_{j_1}A^{i_2}_{j_2}\cdots A^{i_n}_{j_n}, \tag{17} \]

analogous to the pseudoscalar \(G=\frac{1}{8}i\varepsilon_{\mu\nu\sigma\rho}M^{\mu\nu}M^{\sigma\rho}\) for the Lorentz group. Its eigenvalues are equal to \((^9)\)

\[ \tilde C(\mathbf m)=(-1)^{n(n-1)/2}2^n n!\, l_1l_2\cdots l_n. \tag{18} \]

* Here \(r_i\) is the half-sum of the positive roots of the group; note that for all the groups considered \(r_i=(\alpha+1)\varepsilon_i-i\).

In conclusion, we would like to draw the attention of mathematicians to the following unsolved problems:

1) Do formulas (8), (15) generalize to the case of exceptional groups (of type \(G_2, F_4\), etc.), and also to the case of infinite-dimensional unitary representations of classical groups?

2) Along with the operators \(C_p\) defined in (2), symmetrized Casimir operators \(I_p\) occur in applications:

\[ I_p=\frac{1}{p!}\,P\left(A_{i_2}^{i_1}\cdots A_{i_1}^{i_p}\right), \tag{19} \]

where the symbol \(P\) (the symmetrizer) denotes summation over all permutations of the generators standing in parentheses. The operator \(I_p\) is expressed in terms of \(C_q\) with \(q\leq p\); the simplest formulas of this kind (for \(p\leq 5\)) were obtained in \({}^{(6,9)}\). It is desirable to find the relation between \(I_p\) and \(C_p\) in general form, and also to determine whether the eigenvalue of \(I_p\) can be represented in a form analogous to (8) or (9).

The authors express their sincere gratitude to I. M. Gel'fand for discussion of the results of the present work.

Received
30 VI 1966

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