Quiver Hecke algebra module categories of arbitrary symmetrizable type admit Kleshchev stratifications, and quantum-unipotent subcategories are affine highest weight categories.
Cluster Monomials are Dual Canonical
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Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types.
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Affine highest weight structures on module categories over quiver Hecke algebras
Quiver Hecke algebra module categories of arbitrary symmetrizable type admit Kleshchev stratifications, and quantum-unipotent subcategories are affine highest weight categories.