Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2025
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| _version_ | 1866909752899928064 |
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| author | Aoki, Toshitaka Higashitani, Akihiro Iyama, Osamu Kase, Ryoichi Mizuno, Yuya |
| author_facet | Aoki, Toshitaka Higashitani, Akihiro Iyama, Osamu Kase, Ryoichi Mizuno, Yuya |
| contents | The $g$-fan $Σ(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $Σ(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3 Aoki, Toshitaka Higashitani, Akihiro Iyama, Osamu Kase, Ryoichi Mizuno, Yuya Representation Theory Combinatorics 16G20, 52B10 The $g$-fan $Σ(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $Σ(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations. |
| title | Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3 |
| topic | Representation Theory Combinatorics 16G20, 52B10 |
| url | https://arxiv.org/abs/2508.18678 |