Stiefel–Whitney classes of representations of SL(2, 𝑞)

Abstract We describe the Stiefel–Whitney classes (SWCs) of orthogonal representations 𝜋 of the finite special linear groups G = SL ⁡ ( 2 , F q ) G=\operatorname{SL}(2,\mathbb{F}_{q}) , in terms of character values of 𝜋. From this calculation, we can answer interesting questions about SWCs of 𝜋. For instance, we determine the subalgebra of H * ⁢ ( G , Z / 2 ⁢ Z ) H^{*}(G,\mathbb{Z}/2\mathbb{Z}) generated by the SWCs of orthogonal 𝜋, and we also determine which 𝜋 have non-trivial mod 2 Euler class.