Combinatorial Hopf algebras and Towers of Algebras

TL;DR

This paper explores the structure of towers of algebras that produce dual Hopf algebras, establishing a specific dimension formula linking algebra dimensions to combinatorial structures.

Contribution

It applies the construction of dual graded graphs to towers of algebras, deriving a dimension formula for such towers based on their Hopf algebra properties.

Findings

Dimension of algebra towers is r^n n! where r = dim(A_1)

Conditions for towers to produce dual Hopf algebras are characterized

Connection between algebraic structures and combinatorial graphs established

Abstract

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$.