TY - JOUR

T1 - Monoidal categories associated with strata of flag manifolds

AU - Kashiwara, Masaki

AU - Kim, Myungho

AU - Oh, Se jin

AU - Park, Euiyong

N1 - 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.

PY - 2018/4/13

Y1 - 2018/4/13

N2 - We construct a monoidal category Cw,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[Rw,v] of the open Richardson variety associated with w and v. The category Cw,v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, Cw,v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra Aq(n(w))Z[q,q−1] given by Kang–Kashiwara–Kim–Oh. We show that the category Cw,v contains special determinantial modules M(w≤kΛ,v≤kΛ) 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≤kΛ,v≤kΛ). When it is of finite ADE type, we further prove that there is an equivalence of categories between Cw,v and Cu for w,u,v∈W with w=vu and ℓ(w)=ℓ(v)+ℓ(u).

AB - We construct a monoidal category Cw,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[Rw,v] of the open Richardson variety associated with w and v. The category Cw,v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v=id, Cw,v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra Aq(n(w))Z[q,q−1] given by Kang–Kashiwara–Kim–Oh. We show that the category Cw,v contains special determinantial modules M(w≤kΛ,v≤kΛ) 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≤kΛ,v≤kΛ). When it is of finite ADE type, we further prove that there is an equivalence of categories between Cw,v and Cu for w,u,v∈W with w=vu and ℓ(w)=ℓ(v)+ℓ(u).

KW - Categorification

KW - Monoidal category

KW - Quantum cluster algebra

KW - Quiver Hecke algebra

KW - Richardson variety

UR - http://www.scopus.com/inward/record.url?scp=85044083883&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.02.013

DO - 10.1016/j.aim.2018.02.013

M3 - Article

AN - SCOPUS:85044083883

SN - 0001-8708

VL - 328

SP - 959

EP - 1009

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -