The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay

TL;DR

This paper proves that the orbit algebra of a permutation group with polynomially bounded profile is Cohen-Macaulay, providing insights into the structure and generating series of such groups and advancing classification efforts.

Contribution

It establishes that the orbit algebra is Cohen-Macaulay, strengthening previous conjectures and linking group actions with commutative algebra properties.

Findings

Orbit algebra is Cohen-Macaulay.

Generating series is a rational fraction with positive numerator coefficients.

Denominator of the generating series has a combinatorial description.

Abstract

Let $G$ be a group of permutations of a denumerable set $E$. The profile of $G$ is the function $\phi_G$ which counts, for each $n$, the (possibly infinite) number $\phi_G(n)$ 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 $\phi_G(n)$ 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 $\phi_G$ - 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.