Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910287999795200 |
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| author | Douglas, Daniel C. Sun, Zhe |
| author_facet | Douglas, Daniel C. Sun, Zhe |
| contents | For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We show that this basis can be naturally indexed by non-negative integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_01768 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction Douglas, Daniel C. Sun, Zhe Geometric Topology Quantum Algebra 2020: 57M15, 32G15 For a finite-type surface $\mathfrak{S}$, we study a preferred basis for the commutative algebra $\mathbb{C}[\mathscr{R}_{\mathrm{SL}_3(\mathbb{C})}(\mathfrak{S})]$ of regular functions on the $\mathrm{SL}_3(\mathbb{C})$-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface $\mathfrak{S}$. We show that this basis can be naturally indexed by non-negative integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety. |
| title | Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction |
| topic | Geometric Topology Quantum Algebra 2020: 57M15, 32G15 |
| url | https://arxiv.org/abs/2011.01768 |