The cluster complex for cluster Poisson varieties and representations of acyclic quivers
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| Format: | Preprint |
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2023
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| author | Melo, Carolina Chávez, Alfredo Nájera |
| author_facet | Melo, Carolina Chávez, Alfredo Nájera |
| contents | Let $\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $Δ^+(\mathcal{X})$ was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on $\mathcal{X}$ that restrict to a character of a seed torus. Every seed ${ \bf s}$ for $\mathcal{X}$ determines a fan realization $Δ^+_{\bf s}(\mathcal{X})$ of $Δ^+(\mathcal{X})$. For every ${\bf s}$ we provide a simple and explicit description of the cones of $Δ^+_{\bf s}(\mathcal{X})$ and their facets using ${\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $Δ^+_{ \bf s}(\mathcal{X})$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $Δ^+_{\bf s}(\mathcal{X})$ using ${\bf g}$-vectors of (non-necessarily rigid) objects in $\mathsf{K}^{\rm b}(\text{proj} \; kQ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_03626 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The cluster complex for cluster Poisson varieties and representations of acyclic quivers Melo, Carolina Chávez, Alfredo Nájera Representation Theory Algebraic Geometry Combinatorics 13F60 Let $\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $Δ^+(\mathcal{X})$ was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on $\mathcal{X}$ that restrict to a character of a seed torus. Every seed ${ \bf s}$ for $\mathcal{X}$ determines a fan realization $Δ^+_{\bf s}(\mathcal{X})$ of $Δ^+(\mathcal{X})$. For every ${\bf s}$ we provide a simple and explicit description of the cones of $Δ^+_{\bf s}(\mathcal{X})$ and their facets using ${\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $Δ^+_{ \bf s}(\mathcal{X})$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $Δ^+_{\bf s}(\mathcal{X})$ using ${\bf g}$-vectors of (non-necessarily rigid) objects in $\mathsf{K}^{\rm b}(\text{proj} \; kQ)$. |
| title | The cluster complex for cluster Poisson varieties and representations of acyclic quivers |
| topic | Representation Theory Algebraic Geometry Combinatorics 13F60 |
| url | https://arxiv.org/abs/2310.03626 |