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Some relational structures with polynomial growth and their associated algebras

Maurice Pouzet·Nicolas M. Thiéry
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Abstract

The profile of a relational structure R is the function phi_R which counts for every integer n the number, possibly infinite, phi_R(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. Several graded algebras can be associated with R in such a way that the profile of R is simply the Hilbert function. An example of such graded algebra is the age algebra introduced by P.~J.~Cameron. In this paper, we give a closer look at this association, particularly when the relational structure R decomposes into finitely many monomorphic components. In this case, several well-studied graded commutative algebras (e.g. the invariant ring of a finite permutation group, the ring of quasi-symmetric polynomials) are isomorphic to some age algebras. Also, phi_R is a quasi-polynomial, this supporting the conjecture that, with mild assumptions on R, phi_R is a quasi-polynomial when it is bounded by some polynomial.

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Maurice Pouzet, Nicolas M. Thiéry. 2006-01-11. Some relational structures with polynomial growth and their associated algebras. https://arxiv.org/abs/math/0601256

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