Quantum indistinguishability from general representations of $SU(2n)$

A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)] is generalized within a group-theoretical framework. The construction of Berry and Robbins is reformulated in terms of certain locally flat vector bundles over $n$-particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group $SU(2n).$ The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spin-statistics connection is typically broken for general $SU(2n)$ representations, which may admit, for a given value of spin, both Bose and Fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zero-weight representations of a (generalized) Weyl group of $SU(2n).$ A formula for this decomposition is obtained using the Littlewood–Richardson theorem for the decomposition of representations of $U(m+n)$ into representations of $U(m)×U(n).$