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Explore the implementation of non-abelian symmetries in tensor network computations through this 29-minute conference talk from JuliaCon Global 2025. Learn how tensor network methods provide numerical algorithms for studying strongly correlated systems in condensed matter and high energy physics by constructing efficient low-rank approximations of high-dimensional quantum wavefunctions. Discover how the ITensors.jl Julia library offers accessible tensor network algorithms through a high-level interface designed for both experts and non-experts. Understand the critical role of symmetries like translation invariance and SU(2) spin symmetry in physical systems and how they can be encoded directly into tensor networks to enforce constraints and dramatically improve computational performance. Examine the challenges of implementing non-abelian symmetries compared to abelian ones, including the need for complex internal tensor structures and specialized manipulation tools. Delve into representation theory frameworks where tensor network legs are associated with representations of symmetry groups, and symmetric tensors are defined as invariant under group actions. Master the decomposition between multiplicity and structural parts that enables storing only the multiplicity component without information loss, resulting in significant performance gains. Gain detailed insights into the specific algorithms used to encode non-abelian symmetries in tensor networks and their practical implementation within the ITensors.jl software library.
