A Hopf algebraic approach to the theory of group branchings

TL;DR

This paper introduces a Hopf algebraic framework for understanding the representation rings of subgroups of GL(n) that stabilize tensors with Young symmetries, linking algebraic structures with combinatorial methods.

Contribution

It develops a Hopf algebra twist approach to describe the Grothendieck ring of these subgroups, using 2-cocycles and plethysms, and emphasizes a formal algebraic perspective.

Findings

Representation ring described as a Hopf algebra twist

Use of 2-cocycle derived from Cauchy kernel

Incorporation of Schur-Weyl duality and coproducts

Abstract

We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $H_\pi$ of the general linear group 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. A detailed analysis including combinatorial proofs for our results can be found in math-ph/0505037. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions.