Endomorphisms of Equivariant Algebraic $K$-theory

Abstract

We prove that for the action of a finite constant group scheme, equivariant algebraic $K$-theory is represented by a colimit of Grassmannians in the equivariant motivic homotopy category. Using this result we show that the set of endomorphisms of the equivariant motivic space defined by $K_0(G,-)$ coincides with the set of endomorphisms of infinite Grassmannians in the equivariant motivic homotopy category by explicitly computing the equivariant $K$-theory of Grassmannians.