Mullineux involution and twisted affine Lie algebras

We use Naito-Sagaki’s work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395–412, J. Algebra 251 (2002) 461–474] on LakshmibaiSeshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graph of basic representations of twisted affine Lie algebras of type A (2) 2l and of type D (2) l+1. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (resp., the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (resp., in the even case) are obtained. ∗Keyword: Lakshmibai-Seshadri paths, orbit Lie algebras, Mullineux involution. ∗Research supported by the URF of Victoria University of Wellington and the National Natural Science Foundation of China (Project 10401005). The author wishes to thank the School of Mathematics, Statistics and Computer Science at VUW for their hospitality during his visit in 2005.