Monoidal categories associated with strata of flag manifolds

Abstract

We construct a monoidal category ${\mathcal{C}}_{w,v}$ which categorifies the doubly-invariant algebra ${}^{N^{\prime }(w)}\mathbb{C}[N{]}^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\mathbb{C}[{\mathcal{R}}_{w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category ${\mathcal{C}}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v=\mathrm{id}$, ${\mathcal{C}}_{w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w){)}_{\mathbb{Z}[q,q^{-1}]}$ given by Kang-Kashiwara-Kim-Oh. We show that the category ${\mathcal{C}}_{w,v}$ contains special determinantial modules $\mathsf{M}(w_{\le k}\Lambda ,v_{\le k}\Lambda )$ for $k=1,\dots ,\ell (w)$, 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 $\mathsf{M}(w_{\le k}\Lambda ,v_{\le k}\Lambda )$. When it is of finite $ADE$ type, we further prove that there is an equivalence of categories between ${\mathcal{C}}_{w,v}$ and ${\mathcal{C}}_u$ for $w,u,v\in \mathsf{W}$ with $w=vu$ and $\ell (w)=\ell (v)+\ell (u)$.

50 pages, minor revision, to appear in Advances in Mathematics

References in corpus (6)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
• Monoidality of Kato's Reflection Functors
• Affinizations and R-matrices for quiver Hecke algebras
• On the monoidality of Saito reflection functors

Cited by in corpus (2)

• Monoidality of Kato's Reflection Functors
• On the monoidality of Saito reflection functors