The Orbit Algebra of an Oligomorphic Permutation Group with Polynomial Profile is Cohen-Macaulay

Justine Falque ; Nicolas M. Thiéry

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

Let G be a group of permutations of a denumerable set E. The profile of G is the function φ_G which counts, for each n, the (possibly infinite) number φ_G(n) of orbits of G acting on the n-subsets of E. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever φ_G(n) is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the \textbf{orbit algebra} of G - a graded commutative algebra invented by Cameron and whose Hilbert function is φ_G - is finitely generated.

In this paper we announce a proof of a stronger statement: the orbit algebra is Cohen Macaulay; it follows that the generating series of the profile is a rational fraction whose denominator admits a combinatorial description and the numerator is non-negative.

The proof uses classical techniques from actions of permutation groups, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial.