Shifted Littlewood-Richardson SLR coefficient symmetries and involutions

Financiador

Organização

Publicações

A cactus group action on a shifted tableau crystal and a shifted Berenstein-Kirillov group

Publication . Rodrigues, Inês Martins; Azenhas, Olga Maria da Silva; Torres, Maria Manuel Correia

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.

2021-10Tese de doutoramento

Acesso aberto

Ver mais

Entidade financiadora

Fundação para a Ciência e a Tecnologia

Número da atribuição

PD/BD/128414/2017