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1. ROOTS OF CHARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID

   Chekhov, L.; Shapiro, M.; Shibo, H. (July 2022, Uspehi matematičeskih nauk)

   We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the A_n-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A−λA^T)=0, with A∈A_n, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra sl_n. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of A_n-groupoid to a A_{Sp_2m}-groupoid. 

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2. Malnormal matrices

   https://doi.org/10.1090/proc/15821  

   Mulcahy, Garrett; Sinclair, Thomas (July 2022, Proceedings of the American Mathematical Society)

   We exhibit an operator norm bounded, infinite sequence { A n } \{A_n\} of 3 n × 3 n 3n \times 3n complex matrices for which the commutator map X ↦ X A n − A n X X\mapsto XA_n - A_nX is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert–Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and provide numerical evidence. 

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3. Synthesis of Logical Clifford Operators via Symplectic Geometry

   Rengaswamy, Narayanan; Calderbank, Robert; Kadhe, Swanand; Pfister, Henry D. (January 2018, IEEE International Symposium on Information Theory)

   Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in CN ×N as a 2m × 2m binary sym- plectic matrix, where N = 2m. We show that for an [m, m − k] stabilizer code every logical Clifford operator has 2k(k+1)/2 symplectic solutions, and we enumerate them efficiently using symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary symplectic matrices. For a given operator, our assembly of all of its physical realizations enables optimization over them with respect to a suitable metric. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the well- known [6, 4, 2] code. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a. 

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4. On the Whittaker range of the generalized metaplectic theta lift

   https://doi.org/10.1016/j.jnt.2023.08.007  

   Friedberg, Solomon; Ginzburg, David (February 2024, Journal of Number Theory)

   The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to au- tomorphic representations on special orthogonal groups. It is of interest to vary the orthog- onal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r + 1 groups for the lift from the r-fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower. 

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5. An intrinsic approach to relative braid group symmetries on ı$$\imath$$quantum groups

   https://doi.org/10.1112/plms.12562  

   Wang, Weiqiang; Zhang, Weinan (November 2023, Proceedings of the London Mathematical Society)

   Abstract We initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time. 

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