Fans and polytopes in tilting theory I: Foundations
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arXiv
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2022
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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 | For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $Δ(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $Δ(A)$. Using $g$-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano.
We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_15213 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fans and polytopes in tilting theory I: Foundations Aoki, Toshitaka Higashitani, Akihiro Iyama, Osamu Kase, Ryoichi Mizuno, Yuya Representation Theory Combinatorics Category Theory Rings and Algebras 16G20 For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $Δ(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $Δ(A)$. Using $g$-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$. |
| title | Fans and polytopes in tilting theory I: Foundations |
| topic | Representation Theory Combinatorics Category Theory Rings and Algebras 16G20 |
| url | https://arxiv.org/abs/2203.15213 |