On the Positivity Conjecture for Finite Abelian p-Groups

Abstract

For a partition $\underline{\lambda }=({\lambda }_1^{{\rho }_1}>{\lambda }_2^{{\rho }_2}>{\lambda }_3^{{\rho }_3}>\dots >{\lambda }_k^{{\rho }_k})$ and its associated finite $\mathcal{R}$-module ${\mathcal{A}}_{\underline{\lambda }}=\underset{i=1}{\stackrel{k}{\oplus }}(\mathcal{R}/{\pi }^{{\lambda }_i}\mathcal{R}{)}^{{\rho }_i}$, where $\mathcal{R}$ is a discrete valuation ring, with maximal ideal generated by a uniformizing element $\pi$, having finite residue field $\mathbf{k}=\mathcal{R}/\pi \mathcal{R}\cong {\mathbb{F}}_q$, the number of orbits of pairs $n_{\underline{\lambda }}(q)=\mid {\mathcal{G}}_{\underline{\lambda }}\backslash ({\mathcal{A}}_{\underline{\lambda }}\times {\mathcal{A}}_{\underline{\lambda }})\mid$ 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.

60 Pages

References in corpus (1)

• Degenerations and orbits in finite abelian groups