Glider representation rings of finite groups and glider character theory
Frederik Caenepeel, Fred Van Oystaeyen · 2019
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;s theorem.