Sydney University Algebra Seminar

    University of Sydney

    School of Mathematics and Statistics

    Algebra Seminar

    Bob Howlett
    University of Sydney

    Klyachko's model for the complex representations of finite general linear groups.

    Friday 17th September, 12-1pm, 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 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    This is joint work with Charles Zworestine: it appeared in his PhD thesis (1993), but is otherwise unpublished.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.