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Abstract

We introduce and study a filtration on the representation ring R(G) of an affine algebraic group G over a field. This filtration, which we call Chow filtration, is an analogue of the coniveau filtration on the Grothendieck ring of a smooth variety. We compare it with the other known filtrations on R(G) and show that all three define on R(G) the same topology. For any n≥1⁠, we compute the Chow filtration on R(G) for the special orthogonal group G:=O+(2n+1)⁠. In particular, we show that the graded group associated with the filtration is torsion-free. On the other hand, the Chow ring of the classifying space of G over any field of characteristic ≠2 is known to contain non-zero torsion elements. As a consequence, any sufficiently good approximation of the classifying space yields an example of a smooth quasi-projective variety X such that its Chow ring is generated by Chern classes and at the same time contains non-zero elements vanishing under the canonical homomorphism onto the graded ring associated with the coniveau filtration on the Grothendieck ring of X⁠.