The virtual cactus group and Littelmann paths

TL;DR

This paper introduces a virtual cactus group that extends previous work, demonstrating its compatibility with Littelmann path actions and generalizing earlier symplectic cases.

Contribution

It defines a new virtual cactus group and shows its action on Littelmann paths aligns with existing virtualization maps, broadening the group's applicability.

Findings

The virtual cactus group is compatible with Littelmann path actions.

The definition generalizes previous symplectic cactus groups.

The virtualization map maintains group action consistency.

Abstract

We define a virtual cactus group and show that the cactus group action on Littelmann paths is compatible with the virtualization map defined by Pan-Scrimshaw \cite{PS18}. Our definition generalizes the group with the same name defined for the symplectic Lie algebra by the author in joint work with O. Azenhas and M. Tarighat-Feller.