Ad-nilpotent ideals and the Shi arrangement

Authors:
Chao-Ping Dong
Affiliations:
Institute of Mathematics, Hunan University, Changsha 410082, China and School of Mathematics, Shandong University, Jinan 250100, China
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

Citing 4
Cited 0

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Journal of Algebraic Combinatorics: An International Journal
On free deformations of the Braid arrangement

European Journal of Combinatorics
Deformations of Coxeter hyperplane arrangements

Journal of Combinatorial Theory Series A
Enumerative Combinatorics: Volume 1

Enumerative Combinatorics: Volume 1

Quantified Score

Hi-index	0.00

Visualization

Abstract

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the I-deleted Shi arrangement Shi(I) naturally emerges. This arrangement interpolates between the Coxeter arrangement Cox and the Shi arrangement Shi, and breaks the symmetry of Shi in a certain symmetrical way. Among other things, we determine the characteristic polynomial @g(Shi(I),t) of Shi(I) explicitly for A"n"-"1 and C"n. More generally, let Shi(G) be an arbitrary arrangement between Cox and Shi. Armstrong and Rhoades recently gave a formula for @g(Shi(G),t) for A"n"-"1. Inspired by their result, we obtain formulae for @g(Shi(G),t) for B"n, C"n and D"n.