  For an odd prime $p$, we consider the Chern classes $\gamma_i$ of the
conjugation representation of the projective unitary group $PU(p^l)$. We show
that the restriction of $\gamma_i$ on a maximal elementary abelian $p$ subgroup
are expressed as Dickson invariants, a collection of purely algebraically
defined elements in the polynomial ring $\mathbb{F}_p[x_1,\cdots,x_n]$.
Furthermore, we show some relations in the cohomology algebra
$H^*(BPU(p^2);\mathbb{F}_p)$ involving the classes $\gamma_i$.
