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16 February 2025
 
  » arxiv » 2301.01498

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Fans and polytopes in tilting theory II: $g$-fans of rank 2
Toshitaka Aoki ; Akihiro Higashitani ; Osamu Iyama ; Ryoichi Kase ; Yuya Mizuno ;
Date 4 Jan 2023
Abstract$g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. Our second main result gives a necessary and sufficient condition for algebras of rank 2 to be $g$-convex.
Source arXiv, 2301.01498
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