Tagged mapping class groups: Auslander–Reiten translation

Abstract

Abstract. We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case.As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category D(ΓS) associated to the surface and the corresponding Seidel-Thomas braid group of D(ΓS) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions

“…In Section 7, we show that the top element of Ý Ý Ñ F GpT q is obtained by rotating arcs in the bottom element of Ý Ý Ñ F GpT q (see Theorem 7.7). This result recovers one of Brüstle and Qiu (see [8]) in the case where the surface is a disk without punctures.…”

Maximal green sequences for quivers of finite mutation type

“…Independent mutation sequence. Brústle and Qiu showed that it is necessary that a maximal green sequence for a triangulation of a surface with punctures must change the tagging at every puncture [5]. One might hope that we could apply µ cycle to every puncture of a closed marked surface to obtain a maximal green sequence, but that is not the case.…”

“…Example 4.6. Consider the triangulation given in Figure 5 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, α, β}. An independence path for the arc labelled α is shown in the figure in green.…”

Non-leaving-face property for marked surfaces

“…We study in this paper the non-leaving-face property of an exchange graph coming from an unpunctured marked surface (S, M ), where S denotes the surface and M the set of marked points on the boundary of S. This exchange graph, as cluster exchange graph of the cluster algebra associated with (S, M ), or the cluster category (S, M ), has been introduced in [FST08], and since then intensely studied in various papers, see [LF09,BZ11,BZ13,BQ15] and others. However, the question of shortest paths of mutations has not been addressed in this context previously.…”