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

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.

arXiv comment27 pages. arXiv admin note: text overlap with arXiv:1305.5328 by other authors

References in corpus (1)

• Degenerations and orbits in finite abelian groups