webs 的主项 strandings
Leading term strandings for webs
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中文总结 AI 辅助
本文研究量子群基本表示张量积中不变向量对应的 webs 的主项 strandings，给出其顺时针开放 strand 约束、$\frak{sl}_n$ web 基的非递归构造及 $\frak{sl}_3$ web 的 strand 反转关系。
中文摘要 AI 辅助
web 是一种平面图，编码量子群基本表示张量积中的不变向量。$\boldsymbol{\frak{sl}_n}$ web 的 stranding 是一种带色定向曲线系统，记录其编码向量的一个单项式。本文聚焦于识别和构造主项 strandings，即那些记录 web 向量相对于单项式字典序的主项的 strandings。我们证明，主项 stranding 的每个开放 strand 都是顺时针的，这对这类 strandings 的边界数据施加了足够的约束，从而为一组 web 构成 web 基提供了充分判据。我们从行严格表构造一个具有指定主项的 web，所得 web 构成一个 web 基，给出了 Fontaine 的 $\boldsymbol{\frak{sl}_n}$ web 基的非递归构造。对于无平顶点的 $\boldsymbol{\frak{sl}_3}$ web，我们利用 web 面的深度识别其主项 stranding。最后，我们证明，任意 $\boldsymbol{\frak{sl}_3}$ web 的主项 stranding 都可通过一系列称为 strand 反转的操作，从任意 stranding 得到。
英文摘要
A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, which constrains the boundary data of such strandings enough to yield a sufficient criterion for a set of webs to form a web basis. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis, giving a non-recursive construction of Fontaine's $\mathfrak{sl}_n$ web bases. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we show a leading term stranding for any $\mathfrak{sl}_3$ web can be reached from an arbitrary stranding via a sequence of operations called strand reversals.
