Irreducible Basis for Permutation Representations

Abstract

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 ≤n, the representation P is linearized. In general, the representation obtained is reducible, so it is decomposed into irreducible components. Decomposition of L into invariant subspaces is determined by a unitary transformation leading from the basis { │j)} to a new, symmetry adapted or irreducible, basis {T γ)}. This problem is quite generally solved by means of the so-called Sakata matrix. Some possible physical applications are indicated. PACS numbers: 02.20.—a, 03.65.Bz