Séminaire Lotharingien de Combinatoire, B86b (2026), 70 pp.
Chudamani Pranesachar Anil Kumar
On the Positivity Conjecture for Finite Abelian p-Groups
Abstract.
For a partition λ =
(λ1ρ1,
λ2ρ2,
λ3ρ3, ...,
λkρk)
and its associated finite R-module
Aλ = ⨁i=1k (R/πλiR)ρi, where R is a
discrete valuation ring, with maximal ideal generated by a
uniformizing element π, with at
least three elements, having finite residue field k = R/πR
≅ Fq, the number of orbits of pairs
nλ(q) =
| Gλ \ (Aλ ×
Aλ) |
for the diagonal action of the automorphism group
Gλ = Aut(Aλ), is a polynomial in q with integer
coefficients. The positivity conjecture states that these coefficients
are in fact non-negative. In this article, we prove this conjecture.
Received: June 28, 2021.
Revised: November 10, 2023.
Accepted: November 21, 2023.
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