We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these iden- tities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson- Eriksson and Reiner. In other types the identities appear to be new. For type An, the affine colored partitions form another family of combinatorial objects in bijec- tion with (n + 1)-core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmanni- ans in terms of affine partitions and a generalization of Young’s lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted.
