Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395-412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461-474] on Lakshmibai-Seshadri 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 graphs of basic representations of twisted affine Lie algebras of type A_{2 ℓ}⁽²⁾ and of type D_{ℓ + 1}⁽²⁾. 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 (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.