Title : The functor of rational representations for $p$-groups.

Author : Serge Bouc

Abstract : Let $k$ be a field of characteristic $p>0$. Let
$\mathcal{C}_{p,k}$ be the category whose objects are the
finite $p$-groups, morphisms are the $k$-linear combinations
of bisets, and composition of morphisms is obtained by
$k$-linear extension from the usual product of bisets.
Let $\mathcal{F}_{p,k}$ denote the category of $k$-linear
functors from $\mathcal{C}_{p,k}$ to the category of
$k$-vector spaces. 
This paper investigates the structure of the functor $kR_\Q$
mapping a $p$-group $P$ to $kR_\Q(P)=k\otimes_\Z R_\Q(P)$, as
an object of $\mathcal{F}_{p,k}$, where $R_\Q(P)$ is the
Grothendieck group of the category of finite dimensional
$\Q P$-modules. The main result is an explicit description of
the lattice of all subfunctors of $kR_\Q$. In particular for
$p$ odd, it is shown that $kR_\Q$ is a uniserial object in
$\mathcal{F}_{p,k}$. For $p=2$, the lattice of subfunctors of
$kR_\Q$ can be described as the lattice of closed subsets of a
graph whose vertices are the 2-groups of normal 2-rank~1.
In both cases a composition series of $kR_\Q$ is obtained,
which leads to a formula giving the dimension of the evaluations
$S_{Q,k}(P)$ of the simple objects $S_{Q,k}$ of
$\mathcal{F}_{p,k}$ associated to $p$-groups $Q$ of normal
$p$-rank 1, different from $C_p$. This formula can be phrased
in terms of rational representations of $P$, but also in terms
of the geometry of the lattice of subgroups of $P$, using the
notions of basic subgroups and origins. For example, if $p=2$,
the dimension of $S_{\un,k}(P)$ is equal to the number of
absolutely irreducible $\Q P$-modules.