Orbits of pairs in finite modules over discrete valuation rings and permutation representations [HBNI Th72]
Anil Kumar · INFLIBNET · 2014
Let A be a discrete valuation ring whose maximal ideal is generated by a uniformizing element π, and which has a finite residue field Fq. Let Λ denote the set of all sequences of symbols of the form (1) (λ1 1 , λ ρ2 2 , . . . , λ ρk k ), where λ1 > λ2 > . . . > λk is a strictly decreasing sequence of positive integers and ρ1, ρ2, . . . , ρk are positive integers. We allow the case where k = 0, resulting in the empty sequence, which we denote by ∅. Every finite A-module Aλ is, up to isomorphism, of the form (2) Aλ = (A/πλ1A)⊕ρ1 ⊕ (A/πλ2A)⊕ρ2 ⊕ . . .⊕ (A/πλkA)⊕ρk for a unique λ ∈ Λ. Let Gλ denote the automorphism group of Aλ. Fix a λ ∈ Λ, the corresponding finite torsion A-module Aλ and its automorphism group Gλ. The group Gλ acts on Aλ by the diagonal action g · (x1, . . . , xn) = (g(x1), . . . , g(xn)) for xi ∈ Aλ and g ∈ Gλ. In this thesis we study the set of Gλ-orbits in Aλ under the above action for n = 2. We find that the cardinality of each orbit is a polynomial in q with integer coefficients and moreover, given such a polynomial, the number of orbits with that cardinality is a polynomial in q with integer coefficients which does not depend on A, but only on the cardinality of the residue field of A. We use these results to analyze the permutation representation of Gλ on the vector spaces C[O] where O runs over Gλ-orbits in Aλ. We are able to prove that these permutation representations are multiplicity free.