Clifford theory for Mackey algebras

We develop a Clifford theory for Mackey algebras. For simple Mackey functors, using their classification we prove Mackey algebra versions of Cliffordʹs theorem and the Clifford correspondence. Let μR(G) be the Mackey algebra of a finite group G over a commutative unital ring R, and let 1N be the unity of μR(N) where N is a normal subgroup of G. Observing that 1NμR(G)1N is a crossed product of G/N over μR(N), a number of results concerning group graded algebras are extended to the context of Mackey algebras, including Fongʹs theorem, Greenʹs indecomposibility theorem and some reduction and extension techniques for indecomposable Mackey functors.