Algebra & Number Theory 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

DOI: 10.2140/ant.2018.12.1787

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