The Collected Works of Eugene Paul Wigner pp 608-654 | Cite as

# On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups

## Abstract

The present article deals with simply reducible groups which are finite or compact and with their unitary representations (more precisely: with their representations which are in unitary form) in general. A simply reducible group satisfies two criteria. First, all classes are ambivalent, i.e., contain with an element *X* also its reciprocal, *X* ^{−1}. Since the characters of all elements of a class are equal to each other in every representation, it follows that the characters of *X* and *X* ^{−1} are equal. If the representation is in unitary form, it is at once evident that the characters of reciprocals are conjugate complex. It therefore follows from the ambivalent nature of all classes that all characters are real in every representation of a simply reducible group. As a result, every representation is equivalent to the conjugate complex representation.

## Keywords

Representation Space Reducible Group Kronecker Product Time Inversion Orthogonality Relation## Preview

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## References

- 1.G. Frobenius and I. Schur,
*Sitzber. Deut. Akad. Wiss. Berlin*,*Kl. Math. Phys. Tech.*, p. 186 (1906).Google Scholar - 2.Cf. F. D. Murnaghan, “Group Representations,” Chapter 3 ( Baltimore, Maryland, 1938 ).Google Scholar
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- 4.j’ + ½) + (j’ − ½) denotes the sum of the representations j’ + ½ and j’ − ½ (i.e. the reducible representation containing j’ + ½ and j’ − ½ . Cf. F. D. Murnaghan, Reference 2.Google Scholar
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*Am. J. Math.*,**63**, 57, 1941. For the present article, the following definitions and theorems are of importance. An irreducible representation which can be transformed into real form is called an integer representation even if the form in which it is used is not real. An irreducible representation which is equivalent to its conjugate complex but cannot be transformed into real form is called a half integer representation. The representations^{DU)}of the three dimensional rotation group with integer*j*are examples for the former, those with half odd integer*j*,examples for the latter. The Kronecker product of two integer representations or of two half integer representations contains only integer representations; the Kronecker product of an integer and a half integer representation contains only half integer representations. A representation which occurs in the symmetrized part of the Kronecker product of an integer representation with itself, or in the antisymmetrized part of the Kronecker product of a half integer representation with itself, is called an even representation. Representations occurring in the antisymmetrized square of an integer representation, or the symmetrized square of a half integer representation are called odd representations. Both even and odd representations are integer; examples for the even and odd are again the^{DU)}with even and odd*j.*Some representations may not occur in the Kronecker product of any representation with itself; these are then neither even, nor odd. However, no representation can be both even and odd and every integer representation of the three-dimensional rotation group (or of the two-dimensional unitary group) is either even or odd. The identical representation’s*j*is 0; it is an even representation.Google Scholar - 7.
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