The Heisenberg product: from Hopf algebras and species to symmetric functions

TL;DR

This paper introduces the Heisenberg product, a unifying algebraic operation that connects various products and coproducts across species, symmetric functions, and Hopf algebras, simplifying their relationships and extending known results.

Contribution

It defines the Heisenberg product unifying multiple algebraic structures and provides combinatorial formulas for its structure constants, extending previous work.

Findings

The Heisenberg product generalizes classical products in symmetric functions and Hopf algebras.

Explicit combinatorial formulas for structure constants are derived.

The product unifies diverse algebraic operations across different mathematical objects.

Abstract

Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded- product the Heisenberg product, that for the highest and lowest degrees produces the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the "commutative context", and endomorphisms of Hopf algebras in the "non-commutative" environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry -to which we add the Heisenberg coproduct for quasi-symmetric functions-, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasi- symmetric functions, we describe explicitly the new operations in terms of alphabets.