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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 (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, which 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 ([ 33] and [ 29].) 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 [ 12, 17, 29, 33]. 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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This content will become publicly available on July 3, 2026
Web Bases in Degree Two From Hourglass Plabic Graphs
Webs give a diagrammatic calculus for spaces of $$U_{q}(\mathfrak{sl}_{r})$$-tensor invariants, but intrinsic characterizations of web bases are only known in certain cases. Recently, we introduced hourglass plabic graphs to give the first such $$U_{q}(\mathfrak{sl}_{4})$$-web bases. Separately, Fraser introduced a web basis for Plücker degree two representations of arbitrary $$U_{q}(\mathfrak{sl}_{r})$$. Here, we show that Fraser’s basis agrees with that predicted by the hourglass plabic graph framework and give an intrinsic characterization of the resulting webs. A further compelling feature with many applications is that our bases exhibit rotation-invariance. Together with the results of our earlier paper, this implies that hourglass plabic graphs give a uniform description of all known rotation-invariant $$U_{q}(\mathfrak{sl}_{r})$$-web bases. Moreover, this provides a single combinatorial model simultaneously generalizing the Tamari lattice, the alternating sign matrix lattice, and the lattice of plane partitions. As a part of our argument, we develop properties of square faces in arbitrary hourglass plabic graphs, a key step in our program towards general $$U_{q}(\mathfrak{sl}_{r})$$-web bases.
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- Award ID(s):
- 2348843
- PAR ID:
- 10626830
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 13
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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