Specht和$\mathfrak{sl}_3$网基之间的转移阵相对于阴影包容是单三角形的
The transition matrix between the Specht and $\mathfrak{sl}_3$ web bases is unitriangular with respect to shadow containment
摘要：
网是具有边界的可平面图，其描述$\mathfrak{sl}_k$的图解表示范畴中的态射。它们被纽结理论家广泛研究，因为编织映射提供了一种用网络来表达链接图的分类方法，产生了量子不变量，如著名的琼斯多项式。表示论中的一个重要问题是识别不同基之间的关系;基变换矩阵中的系数通常描述组合、代数或几何量（例如，Kazhdan-Lusztig多项式）。通过“扁平化”编织图，网也可以被视为编织群表示的基本元素。在本文中，我们定义了两个新的组合结构的网络：带图和一维投影，阴影，测量网络内的区域的深度。作为一个应用，我们解决了一个开放的猜想，即所谓的Specht基础和网络基础之间的变化的基础，这种网络群表示是单三角的$\mathfrak{sl}_3$ -网络。我们使用带图和阴影来构造一个新的偏序，它是通常偏序的细化。事实上，我们证明了对于$\mathfrak{sl}_2$ -网，我们的新的偏序与作者和其他人研究的网上的tableau偏序一致。我们还证明，虽然新的偏序$\mathfrak{sl}_3$ -网是一个改进的先前研究的tableau秩序，这两个偏序不同意$\mathfrak{sl}_3$。
英文摘要：
Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (like, e.g., Kazhdan-Lusztig polynomials). By "flattening" the braiding maps, webs can also be viewed as the basis elements of a symmetric-group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, that measure depths of regions inside the web. As an application, we resolve an open conjecture that the change-of-basis between the so-called Specht basis and web basis of this symmetric-group representation is unitriangular for $\mathfrak{sl}_3$-webs. We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously-studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.
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