论文标题
$ \ mathrm {sl} _3 $ -webs in表面上的热带fock-goncharov坐标
Tropical Fock-Goncharov coordinates for $\mathrm{SL}_3$-webs on surfaces I: construction
论文作者
论文摘要
对于有限型表面$ \ mathfrak {s} $，我们研究了交换性代数$ \ Mathbb {c} [\ Mathscr {\ Mathscr {r} _ {\ Mathrm {Slrm {sl} _3（\ Mathbb {c}）}（\ Mathakak} { $ \ mathrm {sl} _3（\ Mathbb {c}）$ - 字符品种，由Sikora-Westbury介绍。这些基础元素来自与嵌入在表面$ \ mathfrak {s} $的某些三价图相关的跟踪函数。我们表明，通过非阴性整数坐标可以自然地索引此基础，该坐标由Knutson-Tao Rhombus不等式和Modulo 3一致性条件定义。这些坐标是通过Fock和Goncharov的几何理论与角色品种双重版本的热带点相关的。
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.
