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

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree $2^n-1-n$ of the polynomial algebra $\mathbb{F}_2[x_1,\,\ldots ,\,x_n]$, considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of $\mathrm{GL}_n(\mathbb{F}_2)$. We generalize this result to all finite fields by studying a family of finite quotient rings $\mathbf{R}_{n,k}$, $k\in \mathbb{N}^*$, of $\mathbb{F}_q[x_1,\,\ldots ,\,x_n]$, where each $\mathbf{R}_{n,k}$ is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of $\mathbf{R}_{n,k}$, we also show that the space of indecomposable elements of $\mathbb{F}_q[x_1,\,\ldots ,\,x_n]$ in degree $q^{n-1}-n$ has dimension equal to that of a complex cuspidal representation of $\mathrm{GL}_n(\mathbb{F}_q)$, that is $(q-1)(q^2-1)\cdots (q^{n-1}-1)$.Over the field $\mathbb{F}_2$, we also establish a decomposition of the Steinberg summand of $\mathbf{R}_{n,2}$ into a direct sum of suspensions of Brown–Gitler modules. The module $\mathbf{R}_{n,2}$ can be realized as the mod $2$ cohomlogy of a topological space and the result suggests that the Steinberg summand of this space admits a stable decomposition into a wedge of suspensions of Brown–Gitler spectra

Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree $2^n-1-n$ of the polynomial algebra $\mathbb{F}_2[x_1,\,\ldots ,\,x_n]$, considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of $\mathrm{GL}_n(\mathbb{F}_2)$. We generalize this result to all finite fields by studying a family of finite quotient rings $\mathbf{R}_{n,k}$, $k\in \mathbb{N}^*$, of $\mathbb{F}_q[x_1,\,\ldots ,\,x_n]$, where each $\mathbf{R}_{n,k}$ is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of $\mathbf{R}_{n,k}$, we also show that the space of indecomposable elements of $\mathbb{F}_q[x_1,\,\ldots ,\,x_n]$ in degree $q^{n-1}-n$ has dimension equal to that of a complex cuspidal representation of $\mathrm{GL}_n(\mathbb{F}_q)$, that is $(q-1)(q^2-1)\cdots (q^{n-1}-1)$.Over the field $\mathbb{F}_2$, we also establish a decomposition of the Steinberg summand of $\mathbf{R}_{n,2}$ into a direct sum of suspensions of Brown–Gitler modules. The module $\mathbf{R}_{n,2}$ can be realized as the mod $2$ cohomlogy of a topological space and the result suggests that the Steinberg summand of this space admits a stable decomposition into a wedge of suspensions of Brown–Gitler spectra

LANNES' T FUNCTOR ON INJECTIVE UNSTABLE MODULES AND HARISH-CHANDRA RESTRICTION

In the 1980's, the magic properties of the cohomology of elementary abelian groups as modules over the Steenrod algebra initiated a long lasting interaction between topology and modular representation theory in natural characteristic. The Adams-Gunawardena-Miller theorem in particular, showed that their decomposition is governed by the modular representations of the semi-groups of square matrices. Applying Lannes' T functor on the summands L P := Hom Mn(Fp) (P, H * (F p) n) defines an intriguing construction in representation theory. We show that T(L P) ∼ = L P ⊕ H * V 1 ⊗ L δ(P) , defining a functor δ from F p [M n (F p)]-projectives to F p [M n−1 (F p)]-projectives. We relate this new functor δ to classical constructions in the representation theory of the general linear groups