On the Positivity Conjecture for Finite Abelian p-Groups

For a partition $\underline{\lambda} = (\lambda_{1}^{\rho _1}>\lambda_{2}^{\rho _2}>\lambda_{3}^{\rho _3}>\ldots>\lambda_{k}^{\rho _k})$ and its associated finite $\mathcal{R}$-module $\mathcal{A}_{\underline{\lambda}}=\underset{i=1}{\overset{k}{\oplus}} (\mathcal{R}/\pi^{\lambda_i}\mathcal{R})^{\rho_i}$, where $\mathcal{R}$ is a discrete valuation ring, with maximal ideal generated by a uniformizing element $\pi$, having finite residue field ${\bf k}=\mathcal{R}/\pi\mathcal{R}\cong \mathbb{F}_q$, the number of orbits of pairs $n_{\underline{\lambda}}(q)= \mid \mathcal{G}_{\underline{\lambda}}\backslash \big(\mathcal{A}_{\underline{\lambda}}\times \mathcal{A}_{\underline{\lambda}}\big)\mid$ for the diagonal action of the automorphism group $\mathcal{G}_{\underline{\lambda}}= Aut(\mathcal{A}_{\underline{\lambda}})$, is a polynomial in $q$ with integer coefficients. Positivity conjecture states that these coefficients are in fact non-negative. In this article, we prove this conjecture.