Recoupling coefficients of the symmetric group involving outer plethysms

Abstract

In a series of papers we have examined the properties of certain double coset matrix elements (DCME) in the representations of the symmetric group SN that act as recoupling coefficients for outer products carried out via alternate subgroup sequences. In this paper we examine these same properties using symmetrized outer products in SN, which are also known as outer plethysms. The notions of double coset representative, symbol, and matrix element are extended to this case using the theory of semidirect products and little groups. The recoupling coefficients between bases symmetry adapted with respect to the usual outer product and the outer plethysm are examined in detail. Because of the Weyl–Schur construction of irreducible tensors, the recoupling theory of SN is central to a unified recoupling theory of the general linear group and its subgroups.