Symplectic cacti, virtualization and Berenstein-Kirillov groups

Abstract

We explicitly realize an internal action of the symplectic cactus group, originally defined by Halacheva for any finite dimensional complex reductive Lie algebra, on crystals of Kashiwara-Nakashima tableaux. Our methods include a symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic reversal, and virtualization due to Baker. As an application, we define and study a symplectic version of the Berenstein-Kirillov group and show that it is a quotient of the symplectic cactus group.

The Virtual Cactus Group and Littelmann Paths

“…In this paper we will be concerned with pairs of Dynkin diagrams (X, Y ) related by folding, that is, there is an injection of sets of nodes X → Y which induces an injection of the corresponding Lie algebras g X → g Y as described in [BS17]. The main result and aim of this paper is the "virtualization" of the cactus group J X , as defined by Halacheva in [Hal20], and of its action on g X -crystals, transferring certain results obtained for the case C n → A 2n−1 in [ATFT22] to the more general setup described above. This is carried out in Theorem 4 and Theorem 9.…”

“…It consists in defining a group monomorphism J X → J Y compatible with the action of J X and J Y on g X , respectively g Y -crystals. Moreover, by using the virtualization map on Littelmann paths described by Pan-Scrimshaw in [PS18], instead of the Baker virtualization map used in [ATFT22] for Kashiwara-Nakashima tableaux, we obtain a simple rule to compute the partial Schützenberger-Lusztig involutions of Littelmann paths in g X -crystals in terms of partial Schützenberger-Lusztig involutions of Littelmann paths in g Y -crystals. This is carried out in Theorem 9.…”

On a conjecture of Naito-Sagaki: Littelmann paths and Littlewood-Richardson Sundaram tableaux

“…In this paper we will be concerned with pairs of Dynkin diagrams (X, Y ) related by folding, that is, there is an injection of sets of nodes X ֒→ Y which induces an injection of the corresponding Lie algebras g X ֒→ g Y as described in [BS17]. The main result and aim of this paper is the "virtualization" of the cactus group J X , as defined by Halacheva in [Hal20], and of its action on g X -crystals, transferring certain results obtained for the case C n ֒→ A 2n−1 in [ATFT22] to the more general setup described above. This is carried out in Theorem 2 and Theorem 4.…”

“…We especially thank Olga Azenhas for teaching the author a great deal about the cactus group. The ideas in this paper stemmed from the wish to generalize the results in [ATFT22], which was written as part of the project titled The A, C, shifted Berenstein-Kirillov groups and cacti, in the context of the above mentioned meeting. The author was supported by ICERM for this workshop, was also supported by the grant SONATA NCN UMO-2021/43/D/ST1/02290 and partially supported by the grant MAESTRO NCN UMO-2019/34/A/ST1/00263.…”

“…It consists in defining a group monomorphism J X ֒→ J Y compatible with the action of J X and J Y on g X , respectively g Y -crystals. Moreover, by using the virtualization map on Littelmann paths described by Pan-Scrimshaw in [PS18], instead of the Baker virtualization map used in [ATFT22] for Kashiwara-Nakashima tableaux, we obtain a simple rule to compute the partial Schützenberger-Lusztig involutions of Littelmann paths in g X -crystals in terms of partial Schützenberger-Lusztig involutions of Littelmann paths in g Y -crystals. This is carried out in Theorem 4.…”