🚀 go-pugleaf

In article <ARA.92Apr2004750@camelot.ai.mit.edu> ara@zurich.ai.mit.edu (Allan Adler) writes:
>
>
>Let G be the simple group of order 168. G has an irreducible representation
>of dimension 6 over the field with 2 elements. Let A be the elementary
>abelian group of order 2^6=64 that G acts on via this representation.
>Let H be the semidirect product of G and A. 
>
>Does H have an faithful irreducible complex projective representation of
>degree 8 ?
>
The short answer is yes.  The point, as the original poster probably
surmised, is to use a characteristic 2 version of the Weil representation.

Explicitly, let V denote a vector space of dimension 3 over the field
with 2 elements.  Let C[V] denote the complex group ring of V.  We can
view the dual space $V^*$ as a space of complex functions on V via
the sign character on Z/2Z.  Therefore V^* acts on C[V] by multiplication,
while V acts on C[V] by translation.  The two actions don't quite commute,
but they do give rise to a projective (8-dimensional) complex representation
of V x V^*.  This representation lifts to an actual representation
on the Heisenberg group H (central extension of V x V^* by Z/2Z with
2-cocycle given by the natural pairing (a,a*) x (b,b*) --> a*(b) + b*(a).)
The character of the representation rho is easy to calculate; it is zero
on non-zentral elements, 8 on the identity, and -8 on the other
central element.  Since the order of the group is 128, rho 
is irreducible.  On the other hand, any automorphism of the Z/2Z-vector
space W = V x V^* which preserves the symplectic form on W gives
rise to an automorphism of H, and conjugating by
this automorphism does not change the character of rho.
Thus the elements of Sp(W) intertwine rho with itself.  By Schur's lemma,
the semidirect product of Sp(W) by H admits a projective 8 dimensional
representation (which in fact is an actual representation if you replace
Sp(W) by Mp(W).)  The restriction of the representation to Sp(W) gives
the Steinberg representation.  

To solve the given problem, we realize the 168-group as the Levi component of 
a maximal parabolic subgroup of Sp(W) and restrict.

						-Michael Larsen

P.S.  Has anyone worked out a theory of dual reductive pairs over finite
fields?