Hopf algebras and characters of classical groups

TL;DR

This paper establishes a Hopf algebra structure on the ring of symmetric functions using Schur functions and applies it to analyze characters of classical groups, deriving branching rules and tensor product decompositions.

Contribution

It introduces a natural Hopf algebra framework for symmetric functions and universal characters of classical groups, enabling systematic analysis of their representation theory.

Findings

Hopf algebra structure on symmetric functions is explicitly constructed.

Universal characters of classical groups form Hopf algebras with well-defined operations.

Hopf algebra methods facilitate derivation of branching rules and tensor product decompositions.

Abstract

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.