Questions and conjectures about the modular representation theory of the general linear group GLn(F2) and the Poincar\'e series of unstable modules

This note is devoted to some questions about the representation theory over the finite field $\mathbb{F}_2$ of the general linear groups $\mathbb{GL_n(F_2)}$ and Poincar\'e series of unstable modules. The first draft was describing two conjectures. They were presented during talks made at VIASM in summer 2013. Since then one conjecture has been disproved, the other one has been proved. These results naturally lead to new questions which are going to be discussed. In winter 2013, Nguyen Dang Ho Hai proved the second conjecture, he disproved the first one in spring 2014. Up to now, the proof of the second one depends on a major topological result: the Segal conjecture. This discussion could be extended to an odd prime, but we will not do it here, just a small number of remarks will be made