MiS Preprint Repository

We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $\sf{H}_\pi$ of the general linear group $\sf{GL}(n)$ which stabilize a tensor of Young symmetry $\{\pi\}$. It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to employ the coproduct of the inner multiplication. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions.