Irreducible basis for permutation representations

For a given finite group G its permutation representation P, i.e. an action on an n-element set, is considered. Introducing a vector space L as a set of formal linear combinations of \j >, 1 less than or equal to j less than or equal to n, the representation P is linearized. In general, the represen tation obtained is reducible, so it is decomposed into irreducible componen ts. Decomposition of L into invariant subspaces is determined by a unitary transformation leading from the basis {\j]} to a new, symmetry adapted or i rreducible, basis {\Gamma r gamma]}. This problem is quite generally solved by means of the so-called Sakata matrix. Some possible physical applicatio ns are indicated.