A Hopf laboratory for symmetric functions

TL;DR

This paper explores the structure of symmetric functions through Hopf algebra theory, highlighting algebraic operations, cohomology, and twisted products, and relating them to quantum field theory concepts.

Contribution

It provides an explicit algebraic framework for symmetric functions using Hopf and bi-algebraic structures, including new insights into twisted products and their applications.

Findings

Symmetric functions are analyzed via Hopf algebra structures.

Twisted products include orthogonal and symplectic symmetric function algebras.

Connections to quantum field theory combinatorics are discussed.

Abstract

An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.