Zero cycles on homogeneous varieties

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of \'etale subalgebras of central simple algebras which we construct. This allows us to show $A_0(X) = 0$ for certain classes of homogeneous varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of Panin.