Cluster automorphism group of braid varieties

TL;DR

The paper describes the cluster automorphism group of braid varieties by constructing a square extended exchange matrix $\widehat{B^{u,\bm{\beta}}}$ from the braid-quiver data and proving $\det(\widehat{B^{u,\bm{\beta}}}) = (-1)^{m+f}$ with integral inverse $A^{u,\bm{\beta}}$. This yields an explicit basis for $\ker\widetilde{B}(Q_{u,\bm{\beta}})$ given by the last $f$ columns of $A^{u,\bm{\beta}}$, and shows that the cluster automorphism group is the torus $\text{Aut}(\mathcal{A}(Q_{u,\bm{\beta}})) \cong (\mathbb{C}^{\times})^{f}$ acting on $X_{u,\bm{\beta}}$ with exponents determined by $A^{u,\bm{\beta}}$. The authors implement an inductive argument via Case A and Case B (using a reflection $R_i$) to establish integrality and determinant, and they illustrate the theory with numerous examples leveraging 3D plabic graphs and computational tools. This provides a concrete mechanism to study cluster automorphisms and their geometric action on braid varieties, and raises questions about combinatorial descriptions of the nonzero entries of $A^{u,\bm{\beta}}$ and the full range of possible torus actions.

Abstract

The cluster automorphism group of a cluster variety was defined by Gekhtman--Shapiro--Vainshtein, and later studied by Lam--Speyer. Braid varieties are interesting affine algebraic varieties indexed by positive braid words. It was proved recently that braid varieties are cluster varieties. In this paper, we propose a description of the cluster automorphism group and its action on braid varieties, and compute several examples.

Figures (9)

• Figure 1: A soap film can be thought of as light pink region which can go over or under the horizontal strands of the graph $G_{u,\beta}$. The darker pink color indicates whether a soap film goes over or under. In short, if a soap film is going over the top strand, or under the bottom strands when it reaches the bridges, then a soap film does not change. In other circumstances, a soap film is getting cut.
• Figure 2: Half arrow configuration near a bridge. Each of six red arrows are half arrows, the name comes from the fact that their weights are $\frac{1}{2}$ when constructing the matrix $H^{u,\bm{\beta}}$.
• Figure 3: Soap films imposed on a 3D plabic graph $G_{u,\bm{\beta}}$ associated to a permutation $u=s_4s_3s_4$ and $\bm\beta=(5,4,3,2,1,4,3,\textcolor{blue}{4},2,5,\textcolor{blue}{3},\textcolor{blue}{4},5)$. For example, the soap film $C_{10}$, indicated by the brown region, is the unbounded region with a number 10. One can see that a number 10 propagates to the leftmost boundary of $G_{u,\bm\beta}$ and thus the vertex $10$ is frozen, thus the vertex $10$ is frozen. Similarly, soap film $C_{3}$, colored with a pink color, is closed and bounded, thus the vertex $3$ is mutable.
• Figure 4: The quiver $Q_{u,\beta}$ from Figure \ref{['fig: running ex']} following from $G_{u,\beta}$ and Definition \ref{['def: half arrow quiver cit']}. The green color indicates that the corresponding vertex is mutable, whereas the red color indicates that the corresponding vertex is frozen. The number $j$ written in a vertex indicates its correspondence with a soap film $C_j$, defined in Remark \ref{['rmk: ordering rmk']}. The red arrows are half arrows as appeared in Figure \ref{['fig:half arrow config']}.
• Figure 5: Local picture related to $x$ and $y$

Theorems & Definitions (68)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2: CGGSGLSBS22
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6: FZ02
• Definition 2.7
• Definition 2.8