Webs are certain combinatorial diagrams which encode morphisms between representations of quantum groups. They have famous connections to integrable lattice models and topological link invariants. Web bases, when they exist, provide an effectively computable diagrammatic calculus for these morphism spaces. Recently, we have introduced web bases for sl(4) and the "two column" case of general sl(n), finally extending seminal work of Kuperberg on sl(3) from 1996. The new combinatorial framework involves an extension of Postnikov's plabic graphs with multiple trip permutations. Three extreme families of basis webs can be identified with the lattices of alternating sign matrices, plane partitions, and the Tamari lattice, thereby providing a concrete link between these objects. Joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Jessica Striker.
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