Permutation Representations of the Orbits of the Automorphism Group of a Finite Module over Discrete Valuation Ring

TL;DR

This paper proves that permutation representations arising from the automorphism group acting on transitive subsets of finite torsion modules over a discrete valuation ring are multiplicity free, providing a complete classification of these subsets.

Contribution

It introduces a complete description of transitive subsets under the automorphism group action, establishing the multiplicity free property of the associated permutation representations.

Findings

Permutation representations are multiplicity free.

Complete classification of transitive subsets of $O imes O$.

Automorphism group actions on finite modules are well-understood.

Abstract

Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.