Algebra Seminar

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 g-invariant 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This is joint work with Charles Zworestine: it appeared in his PhD thesis (1993), but is otherwise unpublished.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.