Non-kissing complexes and tau-tilting for gentle algebras
Yann Palu LAMFA, Université Picardie Jules Verne, Amiens http://www.lamfa.u-picardie.fr/palu/ , Vincent Pilaud CNRS、LIX, École Polytechnique, Palaiseau http://www.lix.polytechnique.fr/~pilaud/ and Pierre-Guy Plamondon Université Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de Versailles https://www.imo.universite-paris-saclay.fr/~plamondon/
Abstract
We interpret the support $\tau$ -tilting complex of any gentle bound quiver as the non-kissing 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 $\mathbf{g}$ -vector fan and prove that it is the normal fan of a non-kissing associahedron.
中文速览
温和箭图（gentle bound quiver）的表示理论与组合拓扑之间存在一座桥梁，但两者的精确对应关系此前并不清晰。本文证明：任意温和箭图的支撑 τ-倾斜复形（support τ-tilting complex）与其"开花箭图"（blossoming quiver）上游走的非接吻复形（non-kissing complex）同构，从而把一个代数问题完全翻译成组合语言，并附上了一张代数–组合双语词典。在此基础上，当非接吻复形有限时，文章进一步证明了其递增翻转图是一个同余一致格（congruence-uniform lattice）的 Hasse 图，并构造出以 g-向量扇为法扇的多面体实现——非接吻结合面体（non-kissing associahedron）。这一框架统一并推广了此前针对网格形状、多边形剖分和路径箭图等特殊情形的分散结果，同时为 τ-倾斜有限代数的突变计算提供了一套纯组合操作方法，对代数表示论与多面体组合学都具有重要意义。
原文 arXiv:1707.07574;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1707.07574v4