Localizations for quiver Hecke algebras. III (Q6634457)

This paper is the third of the authors' series of papers on \textit{localizations} for \textit{quiver Hecke algebras}. Let \(R\) be a quiver Hecke algebra associated with a quantum group \(U_{q}(\mathfrak{g})\) and let \(R\)-gmod\ be the category of finite-dimensional graded \(R\)-modules. The monoidal category \(R\)-gmod\ has a rich and interesting structure so that there are various successful applications including monoidal categorification for quantum cluster algebras presented in [\textit{S.-J. Kang} et al., J. Am. Math. Soc. 31, No. 2, 349--426 (2018; Zbl 1460.13039)], where it was shown that the Grothendieck ring \(K(\mathcal{C}_{w})\) of the subcategory \(\mathcal{C}_{w}\) of \(R\)-gmod\ defined for an element \(w\) of the Weyl group \(W\) is isomorphic to the \textit{quantum nilpotent coordinate ring} \(A_{q}(\mathfrak{n}(w))\) associated with \(w\), and that the category \(\mathcal{C}_{w}\) gives a monoidal categorification of \(A_{q}(\mathfrak{n}(w))\) as a quantum cluster algebra when \(\mathfrak{g}\) is symmetric.\N\NIn the previous two papers [Pure Appl. Math. Q. 17, No. 4, 1465--1548 (2021; Zbl 1495.18020); Proc. Lond. Math. Soc. (3) 127, No. 4, 1134--1184 (2023; Zbl 1540.18023)], the authors developed a localization procedure using left braiders and applied to the subcategories \(\mathcal{C}_{w}\). This paper develops a localization procedure by right braiders and constructs the localization \(\widetilde{\mathcal{C}}_{w,v}\) of the category \(\mathcal{C}_{w,v}\) using the right braiders arising from determinant modules, where the subcategory \ was introduced in [\textit{M. Kashiwara} et al., Adv. Math. 328, 959--1009 (2018; Zbl 1437.17005)] for a pair of Weyl group elements \(w,v\) with \(w\geq v\) in the Bruhat order. As its applications, the authors show several interesting properties of the localized category \(\widetilde{\mathcal{C}}_{w,v}\) including the right rigidity.