Stiefel–Whitney Classes for finite symplectic groups

Abstract

Let 𝑞 be an odd prime power, and 𝐺=Sp⁡(2⁢𝑛,𝑞) the finite symplectic group. We give an expression for the total Stiefel–Whitney Classes (SWCs) for orthogonal representations 𝜋 of 𝐺, in terms of character values of 𝜋 at elements of order 2. We give “universal formulas” for the fourth and eighth SWCs. For 𝑛=2, we compute the subring of the mod 2 cohomology generated by the SWCs 𝑤𝑘⁡(𝜋).