Combinatorial Hopf algebras and towers of algebras

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutchap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In joint work with Thomas Lam and Huilan Li, we apply the composition of these constructions to show that if the algebras A₀, A₁, A₂,... form a tower satisfying the aforementioned axioms, then dim(A_n)  =  dim(A₁)ⁿ n!.