Localizations for quiver Hecke algebras III

Let R be a quiver Hecke algebra, and let \(\mathscr {C}_{w,v}\) be the category of finite-dimensional graded R-module categorifying a q-deformation of the doubly-invariant algebra \(^{N'(w)} {\mathbb {C}}[N] ^{N(v)} \). In this paper, we prove that the localization \(\widetilde{\mathscr {C}}_{w,v}\) of the category \(\mathscr {C}_{w,v}\) can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category \(\widetilde{\mathscr {C}}_{w,v} \) including the right rigidity.