 
Summary: GEOMETRY, STEINBERG REPRESENTATIONS
AND COMPLEXITY
J.L. ALPERIN \Lambda 1 and G. MASON y1
\Lambda Mathematics Department, University of Chicago, IL 60637, U.S.A.
y Mathematics Department, University of California at Santa Cruz, CA 95064,
U.S.A.
Group representation theory often relates quite different areas of mathe
matics and we shall give yet another example of this phenomenon. A con
struction from finite geometries will lead us to a new concept in represen
tation theory which we shall then apply to the representation theory of Lie
type groups. This, in turn, will involve ideas from the homological approach
to modular representations. We shall, therefore, cover a spectrum of ideas.
One construction of finite projection planes involves the use of spreads.
Suppose that V is a 2ndimensional vector space over a finite field k of char
acteristic p. A spread S is a collection of ndimensional subspaces whose
(settheoretic) union is all of V but where the intersection of any two mem
bers of the collection is zero. A group of linear transformations of V preserves
the spread S if its elements permute the members of S.
Proposition 1. If E is an elementary abelian 2group of linear transfor
mations of V which preserve S and p = 2 then there is a subgroup F of E
