Parabolic Tamari lattices in linear type \(B\) (Q6103833)

The authors present a first study of parabolic Coxeter-Catalan objects associated with the hyperoctahedral group, i.e., the Coxeter group of type \(B\). Parabolic aligned elements associated with the type-\(B\) Coxeter group were introduced algebraically by \textit{H. Mühle} and \textit{N. Williams} [Electron. J. Comb. 26, No. 4, Research Paper P4.34, 28 p. (2019; Zbl 1453.20051)] for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. The authors introduce a combinatorial model of the elements of the parabolic quotient of the hyperoctahedral group with respect to some type-\(B\) composition using colored sign-symmetric permutations, and then describe a particular order on the relevant inversions to describe the forcing conditions that determine the parabolic aligned elements, and further characterize them using pattern-avoidance. It was proved that these parabolic aligned elements in type \(B\) form a quotient lattice of the weak order on the whole parabolic quotient. This quotient lattice is called the type-\(B\) parabolic Tamari lattice, and has the same lattice-theoretic properties as the \(c\)-Cambrian lattice. This is the main result of this paper.