Abstract
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
$\mathcal {A}_q(\mathfrak {n}(w))$
, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module
$X$
and categorical interpretation of (co)degree of
$[X]$
. As applications of such
$\mathbb {Z}\mspace {1mu}$
-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and
$\Lambda$
-invariants of
$R$
-matrices in the quiver Hecke algebra theory.