  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) $\sim$ = L P $\oplus$ H * V 1
$\otimes$ L $\delta$(P) , defining a functor $\delta$ from F p [M n (F
p)]-projectives to F p [M n--1 (F p)]-projectives. We relate this new functor
$\delta$ to classical constructions in the representation theory of the general
linear groups.
