A Comparison of cluster algebra structures arising from $i$-boxes and Demazure weaves

Abstract

We compare two cluster algebras related to a positive element $\mathtt{b}$ in the braid group of finite $ADE$ type. One is the localized bosonic extension ${\tilde{\mathbb{A}}}_{\mathbb{C}}(\mathtt{b})$ equipped with an initial seed arising from an admissible chain $\mathfrak{C}$ of $i$-boxes, which is deeply connected to monoidal categorification. The other is the coordinate ring $\mathbb{C}[X(\underline{\Delta }\mathbf{i})]$ of the braid variety $X(\underline{\Delta }\mathbf{i})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where $\mathbf{i}$ and $\underline{\Delta }$ are expression sequences of $\mathtt{b}$ and the half twist $\Delta$, respectively. We explicitly construct a Demazure weave ${\mathfrak{W}}_{\underline{\Delta }}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with $\mathbf{i}$, and prove that there exists an algebra isomorphism ${\phi }_{\mathbf{i}}:{\tilde{\mathbb{A}}}_{\mathbb{C}}(\mathtt{b})\to \mathfrak{C}[X(\underline{\Delta }\mathbf{i})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and ${\mathfrak{W}}_{\underline{\Delta }}(\mathfrak{C})$. Moreover, the isomorphism ${\phi }_{\mathbf{i}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{\mathbf{i},k}\in {\tilde{\mathbb{A}}}_{\mathbb{C}}(\mathtt{b})$ to the coordinates $z_k\in \mathfrak{C}[X(\underline{\Delta }\mathbf{i})]$ indexed by the letters of $\mathbf{i}$. As applications, we investigate a connection between Demazure weaves and signed words via the $i$-boxes and interpret the isomorphism ${\phi }_{\mathbf{i}}$ from the viewpoint of monoidal categorification using Hernandez--Leclerc categories.

Introduction has been revised and references have been added ; 43 pages