The ring of stable characters over GL_•(q)

For a fixed prime power q, let GL_•(q) denote the family of groups GL_N(q) for N∈Z_⩾0. In this paper we study the C-algebra of “stable” class functions of GL_•(q), and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple VI-modules in the sense of [14,10]. A second one comes from characters of parabolic representations. The final two, one originally defined in [6] and the other in [4], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [1].