# $SL_4/\mu_2$ and a mod $8$ congruence

This is a continuation of a previous post.  Recall that we wanted to prove the following claim:

Claim.  Let $\rho$ be a representation of $SL_4$ such that no irreducible subrepresentation of $\rho$ descends to $SL_4/\mu_2$.  Then if $\rho$ is self-dual, we have that $$8\mid\dim \rho.$$