Vol. 12, No. 7, 2018

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When are permutation invariants Cohen–Macaulay over all fields?

Ben Blum-Smith and Sophie Marques

Vol. 12 (2018), No. 7, 1787–1821
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.

Keywords
invariant theory, modular invariant theory, henselization, Stanley–Reisner, Cohen–Macaulay, commutative ring, finite group
Mathematical Subject Classification 2010
Primary: 13A50
Secondary: 05E40
Milestones
Received: 26 February 2018
Revised: 16 May 2018
Accepted: 17 June 2018
Published: 27 October 2018
Authors
Ben Blum-Smith
Department of Natural Sciences and Mathematics
Eugene Lang College, the New School for Liberal Arts
New York City, NY
United States
Sophie Marques
Department of Mathematics and Applied Mathematics
University of Cape Town
Cape Town
South Africa