Theory of symmetry classes.

Abstract

Although a great deal of work has gone into construction of the irreducible representations of the symmetric group n (and of the general linear group) a simple, intuitive characterization of the symmetry classes is missing. Relying on a systematic distinction between permutations of variables and permutations of places, we provide two such characterizations, showing that elements belonging to any such symmetry class can be described in one of two ways: (i) as the solutions of explicitly given (though not independent) sets of linear equations or (ii) as linear combinations of "simple" elements of a given symmetry class, a simple element being a generalization to an arbitrary symmetry class of the notion of a decomposable skew-symmetric tensor.