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Abstract：

The theory of characteristic classes of the projective unitary group PU(n) is fundamental yet very mysterious as of today. We will review recent progresses on the study on this topic, and consider the Chern classes γi of the conjugation representation of the projective unitary group PU(p^l), where p is an odd prime. We show that the restriction of γi on a maximal elementary abelian p subgroup are expressed as Dickson invariants, a collection of purely algebraically defined elements in the polynomial ring Fp[x1, · · · , xn]. Furthermore, we present some relations in the cohomology algebra H∗(BPU(p^2); Fp) involving the classes γi.