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Matrix models for quantum permutations Brannan, Michael


A quantum permutation (or magic unitary) is given by a square matrix whose entries are self-adjoint projections acting on a common Hilbert space $H$ with the property that the row and column sums each add up to the identity operator. Quantum permutations are operator-valued analogues of ordinary permutation matrices and they arise naturally in both quantum group theory and also in the study of quantum strategies for certain non-local games. From the perspective of non-local games, it is often of great importance to know whether or not a quantum permutation (possibly satisfying some additional algebraic relations among its entries) admits a matrix model. I.e., can it be realized via operators on a finite-dimensional Hilbert space $H$ In this talk, I will explain how in the case of ``generic'' quantum permutations, matrix models abound. More precisely, the universal unital $\ast$-algebra $A(N)$ generated by the coefficients of an $N\times N$ quantum permutation is always residually finite dimensional (RFD). Our arguments are based on quantum group and subfactor techniques. As an application, we deduce that the II$_1$-factors associated to quantum permutation groups satisfy the Connes Embedding Conjecture. This is joint work with Alex Chirvasitu and Amaury Freslon.

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