0-Hecke Modules, Domino Tableaux, and Type-B Quasisymmetric Functions

We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$-Hecke algebras. We apply this framework in type B to give representation-theoretic interpretations of a number of noteworthy families of type-B quasisymmetric functions. Next, we construct modules of the type-B $0$-Hecke algebra corresponding to type-B analogues of Schur functions and introduce a type-B analogue of Schur Q-functions; we prove that these shifted domino functions expand positively in the type-B peak functions. We define a type-B analogue of the $0$-Hecke--Clifford algebra, and we use this to provide representation-theoretic interpretations for both the type-B peak functions and the shifted domino functions. We consider the modules of this algebra induced from type-B $0$-Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type-B peak functions, for the type-B quasisymmetric characteristics of the restrictions of these modules.