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1. Leibniz International Proceedings in Informatics (LIPIcs):18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023)

   https://doi.org/10.4230/LIPIcs.TQC.2023.10  

   Kretschmer, William (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Fawzi, Omar; Walter, Michael (Ed.)

   We prove that for any n-qubit unitary transformation U and for any r = 2^{o(n / log n)}, there exists a quantum circuit to implement U^{⊗ r} with at most O(4ⁿ) gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case U. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions. 

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2. Globally irreducible Weyl modules for quantum groups

   Garibaldi, Skip; Nakano, Daniel K.; Guralnick, Robert M. (January 2018, Springer Proceedings in Mathematics and Statistics (Geometric and Topological Aspects of the Representation Theory of Finite Groups), Springer-Verlag)

   The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $$E_{8}$$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $$U_{\zeta}({{\mathfrak g}})$$ where $${\mathfrak g}$$ is a complex simple Lie algebra and $$\zeta$$ ranges over roots of unity. 

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3. ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

   https://doi.org/10.1017/fms.2019.50  

   HERZIG, FLORIAN; KOZIOŁ, KAROL; VIGNÉRAS, MARIE-FRANCE (January 2020, Forum of Mathematics, Sigma)

   Suppose that $$\mathbf{G}$$ is a connected reductive group over a finite extension $$F/\mathbb{Q}_{p}$$ and that $$C$$ is a field of characteristic  $$p$$ . We prove that the group $$\mathbf{G}(F)$$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $$C$$ . 

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4. Dimension drop for diagonalizable flows on homogeneous spaces

   https://doi.org/10.3934/jmd.2024012  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2024, Journal of Modern Dynamics)

   Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

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5. Stability in the high-dimensional cohomology of congruence subgroups

   https://doi.org/10.1112/s0010437x20007046  

   Miller, Jeremy; Nagpal, Rohit; Patzt, Peter (April 2020, Compositio Mathematica)

   We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $$\mathbf{SL}_{n}(\mathbb{Z})$$ . This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $$\mathbf{SL}_{n}(K)$$ for $$K$$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $$\mathbf{SL}_{n}(\mathbb{Z})$$ . 

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