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1. Random quantum graphs

   https://doi.org/10.1090/tran/8584  

   Chirvasitu, Alexandru; Wasilewski, Mateusz (April 2022, Transactions of the American Mathematical Society)

   We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples ( X 1 , ⋯ , X d ) (X_1,\cdots ,X_d) of traceless self-adjoint operators in the n × n n\times n matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: 2 ≤ d ≤ n 2 − 3 2\le d\le n^2-3 . Moreover, the automorphism group is generically abelian in the larger parameter range 1 ≤ d ≤ n 2 − 2 1\le d\le n^2-2 . This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of X i X_i ’s (mimicking the Erdős-Rényi G ( n , p ) G(n,p) model) has trivial/abelian automorphism group almost surely. 

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2. Composing arbitrarily many SU(N) fundamentals

   https://doi.org/10.1016/j.nuclphysb.2023.116314  

   Polychronakos, Alexios P; Sfetsos, Konstantinos (September 2023, Nuclear Physics B)

   We compute the multiplicity of the irreducible representations in the decomposition of the tensor product of an arbitrary number n of fundamental representations of SU(N), and we identify a duality in the representation content of this decomposition. Our method utilizes the mapping of the representations of SU(N) to the states of free fermions on the circle, and can be viewed as a random walk on a multidimensional lattice. We also derive the large-n limit and the response of the system to an external non-abelian magnetic field. These results can be used to study the phase properties of non-abelian ferromagnets and to take various scaling limits. 

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3. Cayley Graphs Without a Bounded Eigenbasis

   https://doi.org/10.1093/imrn/rnaa298  

   Sah, Ashwin; Sawhney, Mehtaab; Zhao, Yufei (December 2020, International Mathematics Research Notices)

   Abstract Does every $$n$$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $$O(1/\sqrt{n})$$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $$n$$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $$O(\sqrt{\log n / n})$$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups. 

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4. Non-isomorphism of A∗n,2≤n≤∞, for a non-separable abelian von Neumann algebra A

   https://doi.org/10.1007/s00039-024-00669-8  

   Boutonnet, Rémi; Drimbe, Daniel; Ioana, Adrian; Popa, Sorin (April 2024, Geometric and Functional Analysis)

   Abstract We prove that ifAis a non-separable abelian tracial von Neuman algebra then its free powersA^∗n,2≤n≤∞, are mutually non-isomorphic and with trivial fundamental group,$$\mathcal{F}(A^{*n})=1$$, whenever 2≤n<∞. This settles the non-separable version of the free group factor problem. 

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5. Non-Abelian symmetry-resolved entanglement entropy

   https://doi.org/10.21468/SciPostPhys.17.5.127  

   Bianchi, Eugenio; Dona, Pietro; Kumar, Rishabh (January 2024, SciPost Physics)

   We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) symmetry. 

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