Branching rules, plethysms and Hopf algebras - some surprises

Each of the classical groups GL(n-1), SO(n) and Sp(n), with n even, may be thought of as subroups of GL(n) that preserve some invariant -a vector, a 2nd rank symmetric tensor and a 2nd rank antisymmetric tensor, respectively. In each case the branching rules from GL(n) to the subgroup are determined by certain series of Schur functions defined by means of generating functions or plethysms. The rules for decomposing tensor products for each of the subgroups are well known. It is shown that each may be derived using the outer Hopf algebra of the ring of symmetric functions. Indeed they are determined by the coproduct of the relevant Schur function series. Other subgroups of GL(n) may be defined as those leaving invariant higher rank tensors of specified symmetry. The corresponding branching rules are once again determined by new series of Schur functions defined by means of plethysms, and it is shown that the decomposition of tensor products is again governed by a coproducts of these series. Amongst the surprises are the fact that these new subgroups may be finite or non-reductive.