Combinatorial Theory

The noncrossing partition poset associated to a Coxeter group \(W\) and Coxeter element \(c\) is the interval \([1,c]_T\) in the absolute order on \(W\). We construct a new model of noncrossing partititions for \(W\) of classical affine type, using planar diagrams (affine types \(\widetilde{A}\) and \(\widetilde{C}\) in this paper and affine types \(\widetilde{D}\) and \(\widetilde{B}\) in the sequel). The model in type \(\widetilde{A}\) consists of noncrossing partitions of an annulus. In type \(\widetilde{C}\), the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete \([1,c]_T\) to a lattice by factoring the translations in \([1,c]_T\), but the combinatorics of the planar diagrams leads us to make different choices about how to factor.

Mathematics Subject Classifications: 20F55, 05E16, 20F36

Keywords: Absolute order, affine Coxeter group, annulus, noncrossing partition