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Laurent family of simple modules over quiver Hecke algebras
Published online by Cambridge University Press: 11 September 2024
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.
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- Research Article
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- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
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