Glider representation rings of finite groups and glider character theory

This chapter deals with the definition of glider representation rings of a chain of subgroups G ₀ < G ₁ < … < G _d = G. It is the free Z-module generated by the irreducible glider representations and if G ₀ is Abelian it is possible to equip it with a well-defined multiplication. The chapter contains an explicit description of the glider representation ring R(1 < G) for G finite Abelian and for for the quaternion group G = Q ₈. To deal with non-Abelian groups, a short exact sequence of of KG ^ab-modules is determined which yields links with ordinary representation theory. In the case of groups of nilpotency class 2 we can use these results to distinguish some iscocategorical groups such as D ₈ and Q ₈. The chapter ends with the introduction of glider characters and the proof of a generalization of Artin's theorem.