Ad-nilpotent ideals and the Shi arrangement

作者：Dong, Chao-Ping^*

来源：Journal of Combinatorial Theory - Series A, 2013, 120(8): 2118-2136.

DOI：10.1016/j.jcta.2013.08.004

• Ad-nilpotent ideal
• Shi arrangement
• Quasi-antichain
• Characteristic polynomial

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 chi(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 chi(Shi(G), t) for A(n_1). Inspired by their result, we obtain formulae for chi(Shi(G), t) for B-n, C-n and D-n.

• 出版日期2013-11
• 单位湖南大学; 山东大学