Hopf Algebras, Distributive (Laplace) Pairings and Hash Products: A unified approach to tensor product decompositions of group characters

TL;DR

This paper introduces a unified algebraic framework using Hopf algebras and hash products to analyze tensor product decompositions of group characters, encompassing classical cases as special instances.

Contribution

It develops a general theory of higher derived hash products within bicommutative Hopf algebras, unifying various classical character decomposition methods.

Findings

Classical tensor product decompositions are special cases of higher derived hash products.

The framework applies to symmetric functions and classical groups.

A connection to formal group laws is established in the appendix.

Abstract

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the Appendix we discuss a relation to formal group laws.