The symmetric group: branching rules, products and plethysms for spin representations

Abstract

It is shown that the symmetric group Sn may be usefully embedded in the orthogonal group On, and that this embedding leads directly to an n-independent 'reduced' notation for both the spin and ordinary representations of Sn. Making use of this embedding, together with the properties of Q-functions (or Hall-Littlewood functions), branching rules for On (spin down) Sn are developed and the general rule for the decomposition of spin representations under Sn (spin down) Sn-1 is obtained. Simple methods are given for calculating all possible Kronecker products involving the spin and ordinary representations of Sn and the resolution of Kronecker squares into their symmetric and antisymmetric parts. The spin representations of Sn are systematically classified as to their orthogonal, symplectic or complex characters. The emphasis throughout is on obtaining results that obviate the need for explicit character tables and presenting results in an n-independent manner as much as possible.