| Let G = Sp(2n, Fq ) be the symplectic group over a finite field of order q, where q is odd. We show that the sum of the degrees of the irreducible complex characters of G is equal to the number of symmetric matrices in G. For q ≡ 1 (mod 4), this follows from results of R. Cow. For q ≡ 3(mod 4), we use a twisted Frobenius-Schur indicator of Kawanaka and Matsuyama and extrapolate from the methods of Cow. The main method is a form of the Brauer-Witt-Berman induction theorem. We also prove that the sum of the degrees of the irreducible complex characters of GSp(2n, Fq ) is equal to the number of symmetric matrices in the group. |
