Abstract
The work of C. Bonnafé, M.Geck, L. Iancu and T. Lam shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether two given signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm. Moreover, we show that a particular extension of these relations can describe Garfinkle's equivalence relation on $r$-domino tableaux which is given through the notion of open cycles. With these results we enunciate the conjectures of Bonnafé et al. and provide necessary tools for their proofs.
