Bijective projections on parabolic quotients of affine Weyl groups

Abstract

Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. In [2], Berg et al. described a bijection between $$n$$n-cores with first part equal to $$k$$k and $$(n-1)$$(n-1)-cores with first part less than or equal to $$k$$k, and they interpret this bijection in terms of these other combinatorial models for the quotient of the affine symmetric group by the finite symmetric group. In this paper, we discuss how to generalize the bijection of Berg---Jones---Vazirani to parabolic quotients of affine Weyl groups in type $$C$$C. We develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points. We are thereby able to analyze this bijective projection in the language of various additional combinatorial models developed by Hanusa and Jones [10], such as abaci, core partitions, and canonical reduced expressions in the Coxeter group.