2018/04/10 by Justine Falque, Falque, Justine, Nicolas M. Thiéry +1
Computer Science · Mathematics · #05A15 #05E15 #13A50 #20B07 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1804.03489
openalex publication_date 2018/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.