Picard Groups in Equivariant Algebra and Stable Homotopy Theory

Traditionally, homotopy groups in G-equivariant stable homotopy theory have been graded over RO (G), the real representation ring of G. It is arguably more natural to grade homotopical structures over the Picard group of the equivariant stable homotopy category. Though there is a canonical map of abelian groups RO (G) Pic (Ho (SpG) ) relating the two, this map is neither injective or surjective in general. Fausk, Lewis, and May give an algebraic expression of Pic (Ho (SpG) ) in terms of the Picard group of the Burnside ring A (G), and this work suggests a folklore isomorphism between Pic (A (G) ) and Pic (MackG). We prove the existence of this folklore isomorphism in the setting of finite groups, then leverage our analysis to prove a classification of invertible Mackey functors in the setting of finite abelian groups. As a consequence, we furnish a classification of invertible A (G) -modules again for G a finite abelian group.