[Paper Review] Monoidal categories associated with strata of flag manifolds

Masaki Kashiwara, Myungho Kim+2 more|arXiv (Cornell University)|Aug 15, 2017

Algebraic structures and combinatorial models14 references3 citations

TL;DR

This paper constructs a monoidal category $\mathscr{C}_{w,v}$ as a subcategory of the graded module category of a quiver Hecke algebra, which categorifies the doubly-invariant algebra ${}^{N'(w)}\mathbb{C}[N]^{N(v)}$ and, upon localization, yields the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety. The key result is that $\mathscr{C}_{w,v}$ provides a monoidal categorification of a quantum cluster algebra structure on $\mathbb{C}[\mathcal{R}_{w,v}]$, with determinantial modules forming a quantum monoidal seed.

ABSTRACT

We construct a monoidal category $\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\mathbb{C}[N]^{N(v)}$ 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 $\mathscr{C}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \mathrm{id}$, $\mathscr{C}_{w,v}$ 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 $\mathscr{C}_{w,v}$ contains special determinantial modules $\mathsf{M}(w_{\le k}Λ, v_{\le k}Λ)$ for $k=1, \ldots, \ell(w)$, 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 $\mathsf{M}(w_{\le k}Λ, v_{\le k}Λ)$. When it is of finite $ADE$ type, we further prove that there is an equivalence of categories between $\mathscr{C}_{w,v}$ and $\mathscr{C}_u$ for $w,u,v \in \mathsf{W}$ with $w = vu$ and $\ell(w) = \ell(v) + \ell(u)$.

Motivation & Objective

• To construct a monoidal category $\mathscr{C}_{w,v}$ that categorifies the doubly-invariant algebra ${}^{N'(w)}\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$.
• To show that $\mathscr{C}_{w,v}$, after localization, realizes the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety $\mathcal{R}_{w,v}$.
• To establish that $\mathscr{C}_{w,v}$ contains special determinantial modules $\mathsf{M}(w_{\leq k}\Lambda, v_{\leq k}\Lambda)$ that commute with each other and form a quantum monoidal seed.
• To prove that when the quiver Hecke algebra is symmetric and of finite $ADE$ type, $\mathscr{C}_{w,v}$ is equivalent to $\mathscr{C}_u$ for $w = vu$ with $\ell(w) = \ell(v) + \ell(u)$.

Proposed method

• The category $\mathscr{C}_{w,v}$ is realized as a full subcategory of the graded module category of a quiver Hecke algebra $R$, using cuspidal decompositions and convex preorders from [24].
• Determinantal modules $\mathsf{M}(w_{\leq k}\Lambda, v_{\leq k}\Lambda)$ are constructed as images under a functor $\mathbb{W}$, which intertwines the action of the Weyl group on weights.
• The $R$-matrices between determinantial modules are analyzed using normalized $R$-matrix formulas, with degree formulas derived under symmetric quiver Hecke algebra assumptions.
• The monoidal structure is established via the convolution product and compatibility with the quantum cluster algebra structure, relying on the admissibility of initial seeds.
• The equivalence between $\mathscr{C}_{w,v}$ and $\mathscr{C}_u$ is proven using the condition $\ell(w) = \ell(v) + \ell(u)$ and the symmetric $ADE$ type of the quiver Hecke algebra.
• The categorification is verified by showing that the image of the initial seed under $\mathbb{W}$ satisfies the admissibility condition for quantum cluster algebras, ensuring a monoidal categorification.

Experimental results

Research questions

• RQ1 How can a monoidal category be constructed to categorify the doubly-invariant algebra ${}^{N'(w)}\mathbb{C}[N]^{N(v)}$ for Weyl group elements $w$ and $v$?
• RQ2 What is the relationship between the category $\mathscr{C}_{w,v}$ and the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety $\mathcal{R}_{w,v}$?
• RQ3 Do the determinantial modules $\mathsf{M}(w_{\leq k}\Lambda, v_{\leq k}\Lambda)$ in $\mathscr{C}_{w,v}$ commute with each other and form a quantum monoidal seed?
• RQ4 Under what conditions does $\mathscr{C}_{w,v}$ become equivalent to $\mathscr{C}_u$ for $w = vu$ with $\ell(w) = \ell(v) + \ell(u)$?
• RQ5 Can the category $\mathscr{C}_{w,v}$ provide a monoidal categorification of a quantum cluster algebra structure on $\mathbb{C}[\mathcal{R}_{w,v}]$?

Key findings

• The category $\mathscr{C}_{w,v}$ is constructed as a subcategory of the graded module category of a quiver Hecke algebra $R$, and it categorifies the doubly-invariant algebra ${}^{N'(w)}\mathbb{C}[N]^{N(v)}$.
• After localization, $\mathscr{C}_{w,v}$ realizes the coordinate algebra $\mathbb{C}[\mathcal{R}_{w,v}]$ of the open Richardson variety $\mathcal{R}_{w,v}$, linking geometric and algebraic structures.
• The category $\mathscr{C}_{w,v}$ contains special determinantial modules $\mathsf{M}(w_{\leq k}\Lambda, v_{\leq k}\Lambda)$ for $k=1,\dots,\ell(w)$, which pairwise commute under the convolution product.
• When the quiver Hecke algebra $R$ is symmetric, a formula for the degree of $R$-matrices between determinantial modules is derived, enabling explicit computation of quantum $R$-matrix actions.
• For finite $ADE$ type quiver Hecke algebras, an equivalence of categories $\mathscr{C}_{w,v} \simeq \mathscr{C}_u$ is established when $w = vu$ and $\ell(w) = \ell(v) + \ell(u)$, reflecting the decomposition of the Weyl group element.
• The pair $\bigl{(}\{\mathsf{M}(w_{\leq p+k}\Lambda_{i_{p+k}}, v_{\leq p+k}\Lambda_{i_{p+k}})\}_{1\leq k\leq q}, \widetilde{B}\bigr{)}$ forms an admissible initial seed, confirming that $\mathscr{C}_{w,v}$ is a monoidal categorification of the quantum cluster algebra $A_{w,v}$.