Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras ⨁n≥0An 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
⨁n≥0An gives rise to graded dual Hopf algebras then we must
have dim(An)=rnn! where r=dim(A1).Comment: 7 page