
School of Mathematics and Statistics
Bob Howlett
University of Sydney
Klyachko's model for the complex representations
of finite general linear groups.
Friday 17th September, 121pm, Carslaw 375.
Let V be a finite dimensional vector space over
F, the field with q elements. Given a
linear transformation g on V, we say that
a bilinear form f on V is "symmetric
modulo g" if
f(x,y) = f(gy,x)
for all x and y. We obtain a formula for
the number of such forms as a sum over ginvariant
subspaces U of the number of symmetric bilinear
forms on U multiplied by a number determined by the
action of g on V/U, and show that this
formula is essentially equivalent to a theorem of
A. A. Klyachko, which states that a certain sum of
induced (complex) characters of the general linear group
on V contains each irreducible character with
multiplicity exactly one.
This is joint work with Charles Zworestine: it appeared in
his PhD thesis (1993), but is otherwise unpublished.
The proof has been very slightly streamlined recently.
