The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay
Abstract
Let G be a group of permutations of a denumerable set E. The profile of G is the function φG which counts, for each n, the (possibly infinite) number φG(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 φG(n) is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of G - a graded commutative algebra invented by Cameron and whose Hilbert function is φG - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial.