Affine Partitions and Affine Grassmannians
Billey, Sara C.
Mitchell, Stephen C.
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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 identities 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 bijection with n + 1 core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the aﬃne Grassmannians 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.