Laurent family of simple modules over quiver Hecke algebra

TL;DR

This paper introduces Laurent families of simple modules over quiver Hecke algebras, demonstrating their role analogous to clusters in quantum cluster algebra theory and revealing positivity phenomena in categorification.

Contribution

It defines quasi-Laurent and Laurent families of simple modules, proves their cluster-like properties, and explores their applications in categorification and R-matrix invariants.

Findings

Laurent families exhibit quantum Laurent positivity.

Categorifying GLS-clusters yields Laurent families.

New skew symmetric pairings on simple modules are introduced.

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 the quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon for 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 GLS-clusters are Laurent families by using the PBW-decomposition vector of a simple module $X$ and categorical interpretation of (co-)degree of $[X]$. As applications of such $\mathbb{Z}$-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.