Some relational structures with polynomial growth and their associated algebras II. Finite generation.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated