Séminaire Lotharingien de Combinatoire, 80B.83 (2018), 12 pp.
Justine Falque and Nicolas M. Thiéry
The Orbit Algebra of an Oligomorphic 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 \textbf{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; 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.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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