Séminaire Lotharingien de Combinatoire, 86B.11 (2022), 12 pp.
Fern Gossow and Oded Yacobi
On the Action of the Long Cycle on the Kazhdan-Lusztig Basis
Abstract.
The complex irreducible representations of the symmetric group carry an important canonical basis called the Kazhdan--Lusztig basis. Although it is difficult to express how general permutations act on this basis, some distinguished permutations have beautiful descriptions. In 2010 Rhoades showed that the long cycle (1,2,...,n) acts by the jeu-de-taquin promotion operator in the case when the irreducible representation is indexed by a rectangular partition. We prove a generalisation of this theorem in two directions: on the one hand we lift the restriction on the shape of the partition, and on the other hand we enlarge the result to the collection of all separable permutations.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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