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Explore a cutting-edge lecture on algebraic combinatorics and quantum group representations presented at IPAM's Integrability and Algebraic Combinatorics Workshop. Delve into Joshua Swanson's groundbreaking research on unifying lattices through hourglass plabic graphs. Discover how web bases provide an effectively computable diagrammatic calculus for morphism spaces in quantum group representations. Learn about the recent introduction of web bases for sl(4) and the "two column" case of general sl(n), extending Kuperberg's seminal work on sl(3) from 1996. Explore the new combinatorial framework involving an extension of Postnikov's plabic graphs with multiple trip permutations. Understand the connections between three extreme families of basis webs and the lattices of alternating sign matrices, plane partitions, and the Tamari lattice. Gain insights into the concrete links between these objects and their implications for integrable lattice models and topological link invariants.
