When are permutation invariants Cohen-Macaulay over all fields?
Abstract
We prove that the polynomial invariants of a permutation group are Cohen-Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The "if" direction of the argument uses Stanley-Reisner theory and a recent result of Christian Lange in orbifold theory. The "only-if" direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen-Macaulayness.
Cite
@article{arxiv.1802.06735,
title = {When are permutation invariants Cohen-Macaulay over all fields?},
author = {Ben Blum-Smith and Sophie Marques},
journal= {arXiv preprint arXiv:1802.06735},
year = {2018}
}
Comments
30 pages, 5 figures. Rewrote the statement and proof of Lemma 3.7 in response to a referee report, and correspondingly updated later material relying on Lemma 3.7 such as Proposition 3.11 (formerly 3.8) statement and proof, and the proof of Proposition 4.2. Also, added a few remarks after the new proof of Lemma 3.7