Laurent family of simple modules over quiver Hecke algebras

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring, coming from the categorification. Then we show that the families of simple modules categorifying Geiβ-Leclerc-Schröer (GLS) clusters are Laurent families by using the Poincaré-Birkhoff-Witt (PBW) decomposition vector of a simple module and categorical interpretation of (co)degree of. As applications of such -vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and -invariants of -matrices in the quiver Hecke algebra theory.