Mullineux involution and twisted affine Lie algebras

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We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, 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 A₂ℓ⁽²⁾ and of type Dℓ₊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 (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.