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• W3036005986 abstract "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 $mathfrak{sl}_3$." @default.
• W3036005986 created "2020-06-25" @default.
• W3036005986 creator A5028867644 @default.
• W3036005986 creator A5037445253 @default.
• W3036005986 date "2020-06-16" @default.
• W3036005986 modified "2023-09-27" @default.
• W3036005986 title "The transition matrix between the Specht and $mathfrak{sl}_3$ web bases is unitriangular with respect to shadow containment" @default.
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• W3036005986 hasPublicationYear "2020" @default.
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