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- Title
- Laurent family of simple modules over quiver Hecke algebras
- KIAS Author
- Kashiwara, Masaki
- Journal
- COMPOSITIO MATHEMATICA, 2024
- Archive
-
- 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 A(q)(n(w)), coming from the categorification. Then we show that the families of simple modules categorifying Gei ss-Leclerc-Schr & ouml;er (GLS) clusters are Laurent families by using the Poincar & eacute;-Birkhoff-Witt (PBW) decomposition vector of a simple module X and categorical interpretation of (co)degree of [X]. As applications of such 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.
