Non-kissing complexes and tau-tilting for gentle algebras

Abstract

We interpret the support τ -tilting complex of any gentle bound quiver as the nonkissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g-vector fan and prove that it is the normal fan of a non-kissing associahedron. ContentsPart 2. The non-kissing complex 2.1. Blossoming quivers 2.1.1. Blossoming quiver of a gentle bound quiver 2.1.2. Dissection and grid bound quivers 2.2. The non-kissing complex 2.2.1. The non-kissing complex 2.2.2. Distinguished walks, arrows and substrings 2.2.3. Flips 2.3. Non-kissing complexes versus support τ -tilting complexes 2.3.1. From walks to strings 2.3.2. From walks to AR-quivers The three authors were partially supported by the French ANR grant SC3A (15 CE40 0004 01).

Scattering Amplitudes and Simple Canonical Forms for Simple Polytopes

“…Instead, a notion of compatibility with a reference quadrangulation Q is used to select a set of quadrangulations with the appropriate poset structure, corresponding to the Stokes polytope, as introduced in [9]. The properties of Stokes polytopes have been studied by mathematicians in [10], [11], and used by physicists to describe the planar φ 4 amplitudes in [5], [8].…”

On Extensions for Gentle Algebras

“…The associated string combinatorics governs the representation theory of gentle algebras, examples of this are the classification of morphisms between string and band modules [10,17] and a characterisation of almost split sequences in terms of string combinatorics [7]. Over last few years, interest in gentle algebras has intensified with many new results appearing, an example of this is the very recent work [20], where string combinatorics is used to classify support τ -tilting modules.…”

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

“…Recenlty, gentle algebras have been associated to triangulations or dissections of surfaces, in connection with cluster algebras [ABCJP10], with ribbon graphs [Sch15,Sch18] or with Fukaya categories of surfaces [HKK17]. It turns out that any gentle algebra can be obtained from a dissection of a surface; this has led to geometric models for their module categories [BCS18] (building on [CSP16]) and τ -tilting theory [PPP18] (see also [BDM + 17,PPP17]). From the point of view of homological algebra, the class of gentle algebras is of particular interest, since it is closed under derived equivalence [SZ03].…”