Affine partitions and affine Grassmannians

TL;DR

This paper establishes a bijection between affine colored partitions and elements of affine Weyl groups, deriving new partition identities and characterizing rationally smooth Schubert varieties in affine Grassmannians.

Contribution

It introduces a novel combinatorial correspondence linking affine partitions with affine Weyl group elements and characterizes smooth Schubert varieties using these partitions.

Findings

New partition identities in certain types

Bijection between affine partitions and affine Weyl group elements

Characterization of rationally smooth Schubert varieties

Abstract

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 $A_{n}$, 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.