## Abstract

We construct a monoidal category C_{w,v} which categorifies the doubly-invariant algebra CN^{′}(w)[N]^{N(v)} associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra C[R_{w,v}] of the open Richardson variety associated with w and v. The category C_{w,v} is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, C_{w,v} is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A_{q}(n(w))_{Z[q,q−1]} given by Kang–Kashiwara–Kim–Oh. We show that the category C_{w,v} contains special determinantial modules M(w_{≤k}Λ,v_{≤k}Λ) for k=1,…,ℓ(w), which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M(w_{≤k}Λ,v_{≤k}Λ). When it is of finite ADE type, we further prove that there is an equivalence of categories between C_{w,v} and C_{u} for w,u,v∈W with w=vu and ℓ(w)=ℓ(v)+ℓ(u).

Original language | English |
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Pages (from-to) | 959-1009 |

Number of pages | 51 |

Journal | Advances in Mathematics |

Volume | 328 |

DOIs | |

State | Published - 13 Apr 2018 |

### Bibliographical note

Funding Information:The research of E.P. was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2017R1A1A1A05001058).

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Categorification
- Monoidal category
- Quantum cluster algebra
- Quiver Hecke algebra
- Richardson variety