Module-Theoretic Approach to the Green Correspondence for Mackey Functors

Abstract

There is a close relationship between Mackey functors and group algebras due to the definition of Mackey functors. Therefore, we study the Green correspondence for Mackey functors and we offer an exposition of Sasaki's characterization, coupled with other perspective on the Mackey functor correspondence using the theory of group algebra . The group algebra's concept of "Green correspondence" asserts that there is a bijective relationship between the isomorphism classes of finitely generated, indecomposable KG-modules having vertex Q and the corresponding isomorphism classes of finitely generated, indecomposable KF-modules also having vertex Q. Here, K represents a commutative ring with a unity element, F is a subgroup of G, and F contains the normalizer of Q in, denoted as N_G (Q). This theory is an essential theory in group representation theory proved by J.A. Green in 1964, and after Green introduced the Green functors (is also called G-functors) to the functor theory, he proved the correspondence for G-functors in 1971. Sasaki adapted this correspondent theorem of Green functors to Mackey functors using terms of functor theory in 1982 and the correspondence was demonstrated for an indecomposable Mackey functor M by utilizing the endomorphism Green functor End(M) and establishing the vertex of M based on the vertex of End(M). On the other hand, there is another aproach to this correspondence using theories and terms in module theory. We provide evidence for the theorem using a similar approach to the proof used for group algebras; our characterization aligns more closely with Green's original module-theoretic method.