By C Chevalley

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35) Note that the last term on the right-hand side where x = ala23la2a13 vanishes identically. 36) 1 These results have been established for the case m33 = 0, the unimodular case. We can obtain the matrix elements for U(3) states simply by replacing all mi; in the above expressions by mi; - rn33, m33 0 O. In principle the method just illustrated for SU(3) and U(3) can be general- ized immediately to arbitrary U(n). There is no inherent limitation to the method whereby normalized states for U(n) in highest weight in the U(n - 1) subgroup are written out immediately in terms of Gel'fand basis functions by means of boson creation operators (and hook patterns) and then lowered by 42 W.

35) are indeed the primitive characters of Si,. The proof involves the introduction of a new set of n variables rj, with its power sums Pk = J= qt and the construction of the sum G. ) (Q1P1)vi . (v) . 977] 0) . over the equivalence classes (v) of S,. , EAn) zi`l7Ai+n-1' 1 [A] 7)A2+n-2, . , 7JAn), . ) l . where, as before, the sum is taken over all partitions [A] with 2n t= 1 A, = p. 35) is then shown to become simply the statement of the orthogonality theorem for primitive characters of S. The details of the proof are available in a large number of standard texts, such as those of Weyl (8) and Hamermesh (4), as well as in Frobenius' original paper (11), to which we refer the reader.

37) Here, ai = 1/i! det Z1. 38) Also, the ith coefficient of x in the expansion of (f (x)) - 1 is given by the permanent of Zi divided by r !. , 1]), which is equal to + 1 for all elements of Si. Hence we have the result: If the eigenvalues of u E U(n) are given by the variables El, ... , en, then all the primitive characters of U(n) which belong to the class of the element u are just the Schur functions o f E1, ... 35), we find that we can write the characters tpPA'(u) in the form ,40,+n-l, EA2 + n - 2 Q En-1 EAnI , E n-2 ,..