HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Conference papers

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

Justine Falque 1 Nicolas M. Thiéry 1
1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
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.
Complete list of metadata

Cited literature [13 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01763795
Contributor : Justine Falque Connect in order to contact the contributor
Submitted on : Wednesday, April 11, 2018 - 3:23:00 PM
Last modification on : Thursday, July 8, 2021 - 3:50:37 AM

Identifiers

  • HAL Id : hal-01763795, version 1
  • ARXIV : 1804.03489

Citation

Justine Falque, Nicolas M. Thiéry. The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay. 30th International Conference of Formal Power Series and Algebraic Combinatorics (FPSAC 2018), Jul 2018, Hanover, United States. ⟨hal-01763795⟩

Share

Metrics

Record views

92

Files downloads

179