Author(s):
Rodrigues, Inês Martins
Date: 2021
Persistent ID: http://hdl.handle.net/10451/51919
Origin: Repositório da Universidade de Lisboa
Project/scholarship:
info:eu-repo/grantAgreement/FCT/OE/PD%2FBD%2F128414%2F2017/PT;
Subject(s): Tableaux desviados; grafos de cristais; involução de Schützenberger; grupo cactus; grupo de Berenstein–Kirillov; Shifted tableaux; crystal graphs; Schützenberger involution; cactus group; Berenstein–Kirillov group; Domínio/Área Científica::Ciências Naturais::Matemáticas
Description
Gillespie, Levinson and Purbhoo recently introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. This structure may be regarded as a directed acyclic weighted graph, with coloured double edges, having vertices the shifted semistandard tableaux. It decomposes into connected components, each one having unique source vertex, whose weight is a strict partition, and sink vertex, with reverse weight. The character of each connected component is the Schur Q-function indexed by the said strict partition. Following a similar approach as Halacheva, for crystals of finite-dimensional representations of the quantized universal enveloping algebra of a finite-dimensional complex reductive Lie algebra, we exhibit a natural internal action of the n-fruit cactus group on the shifted tableau crystal, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet. This includes the shifted crystal reflection operators, which agree with the restrictions of the shifted Schützenberger involution to single-coloured connected components, but unlike the case for type A crystals, these do not need to satisfy the braid relations of the symmetric group. In addition, we define a shifted version of the Berenstein–Kirillov group, by considering shifted Bender–Knuth involutions. Paralleling the works of Halacheva and Chmutov, Glick and Pylyavskyy for type A semistandard tableaux of straight shape, we exhibit another occurrence of the cactus group action on shifted tableau crystals of straight shape via the action of the shifted Berenstein–Kirillov group. We also conclude that the shifted Berenstein–Kirillov group is isomorphic to a quotient of the cactus group. Not all known relations that hold in the classic Berenstein–Kirillov group need to be satisfied by the shifted Bender–Knuth involutions, but the ones implying the relations of the cactus group are verified, thus we have another presentation for the cactus group in terms of shifted Bender–Knuth involutions. We also use the shifted growth diagrams due to Thomas and Yong, together with the semistandardization process of Pechenik and Yong, to provide an alternative proof concerning the mentioned cactus group action.
