Substitutional symmetries and extra degeneracies of real symmetric matrices

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Abstract
The relation between substitutional symmetry operations that leave a real symmetric matrix invariant and the degeneracies exhibited by the matrix in diagonal form are examined. The usual application of group theory to this problem is formulated. Substitutional (and other) symmetries can exist, which do not form part of the invariance group. These symmetries can cause extra degeneracy of the root system and are frequently encountered in some physical applications such as the Hückel model of molecular bonds. Some general features that lead to extra degeneracy are noted and illustrative examples are given for systems of six equivalent centers.

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