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1. Finite-dimensional approximations of certain amalgamated free products of groups

   https://doi.org/10.4171/GGD/826  

   Schafhauser, Christopher (November 2024, Groups, Geometry, and Dynamics)

   A group is called matricial field (MF) if it admits finite-dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of MF groups by showing that given two amenable groups with a common normal subgroup, the amalgamated free product is MF. 

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2. A reciprocal branching problem for automorphic representations and global Vogan packets

   https://doi.org/10.1515/crelle-2019-0016  

   Jiang, Dihua; Liu, Baiying; Xu, Bin (August 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G ,determine the occurrence of an irreducible representation σ of H in the restriction of π to H . The reciprocal branching problem of this classical branching problemis to ask: For an irreducible representation σ of H , find an irreducible representation π of G such that σ occurs in the restrictionof π to H . For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture.In this paper, we investigate the reciprocal branching problem for automorphic representationsof special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied toother classical groups as well. 

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3. Quantum codes and irreducible products of characters

   https://doi.org/10.1007/s10623-025-01599-8  

   Kubischta, Eric; Teixeira, Ian (April 2025, Designs, Codes and Cryptography)

   Abstract In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates. Transversal gates are of significant interest to the quantum information community for their central role in fault tolerant quantum computing. Most unitary$$\text {t}$$ $\text{t}$-designs have never been realized as the transversal gate group of a quantum code. We, for the first time, find nontrivial quantum codes realizing nearly every finite group which is a unitary 2-design or better as the transversal gate group of some error-detecting quantum code. 

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4. Unsupervised Co-Learning on G-Manifolds Across Irreducible Representations

   Fan, Yifeng; Gao, Tingran; Zhao, Zhizhen (January 2020, Advances in neural information processing systems)

   We introduce a novel co-learning paradigm for manifolds naturally admitting an action of a transformation group , motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common base manifold, which provides multiple views for the geometry of the underlying manifold. The consistency across these fibre bundles provide a common base for performing unsupervised manifold co-learning through the redundancy created artificially across irreducible representations of the transformation group. We demonstrate the efficacy of our proposed algorithmic paradigm through drastically improved robust nearest neighbor identification in cryo-electron microscopy image analysis and the clustering accuracy in community detection. 

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5. 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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