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1. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel (January 2023, Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. The linear representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$that are subrepresentations of a symmetric one. 

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2. Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

   https://doi.org/10.1090/cams/19  

   Edie-Michell, Cain (May 2023, Communications of the American Mathematical Society)

   This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type D D - D D case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . This will essentially finish the type I I II classification for s l n \mathfrak {sl}_n modulo type I I classification. When paired with Gannon’s type I I classification for r ≤ 6 r\leq 6 , our results will complete the type I I II classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . 

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3. Random walks on direct products of groups

   https://doi.org/10.4171/JEMS/1550  

   Golsefidy, Alireza Salehi; Srinivas, Srivatsa (November 2024, Journal of the European Mathematical Society)

   Suppose\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})is generated by a symmetric setSof cardinalitynwherepis a prime number. Suppose the Cheeger constants of the Cayley graphs of\operatorname{SL}_{2}(\mathbb{F}_{p})with respect to\pi_{L}(S)and\pi_{R}(S)are at leastc_{0}, where\pi_{L}and\pi_{R}are the projections to the left and the right components of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}), respectively. Then the Cheeger constant of the Cayley graph of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})with respect toSis at leastcwherecis a positive number which only depends onnandc_{0}. This gives an affirmative answer to a question of Lindenstrauss and Varjú. 

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4. A symmetrization inequality shorn of symmetry

   https://doi.org/10.1090/tran/8145  

   Christ, Michael; Maldague, Dominique (August 2020, Transactions of the American Mathematical Society)

   An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space R d \mathbb {R}^d of specified Lebesgue measures, (tuples of) balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For d > 1 d>1 , this inequality only applies to functionals invariant under a diagonal action of Sl ⁡ ( d ) \operatorname {Sl}(d) . We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which Sl ⁡ ( d ) \operatorname {Sl}(d) invariance does not hold. Assuming a more limited symmetry encompassing dilations but not rotations, we show under natural hypotheses that maximizers exist, and, moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the Sl ⁡ ( d ) \operatorname {Sl}(d) –invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that in the absence of partial symmetry, maximizers fail to exist for certain arbitrarily small perturbations of Sl ⁡ ( d ) \operatorname {Sl}(d) –invariant structures. 

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5. Borel and Volume Classes for Dense Representations of Discrete Groups

   https://doi.org/10.1093/imrn/rnab078  

   Farre, James (April 2021, International Mathematics Research Notices)

   Abstract We show that the bounded Borel class of any dense representation $$\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $$G$$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $$\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$$ is uniformly separated in semi-norm from any other representation $$\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$$ for which there is a subgroup $$H\le G$$ on which $$\rho $$ is still dense but $$\rho ^{\prime}$$ is discrete or indiscrete but stabilizes a point, line, or plane in $${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of $${\operatorname{PSL}}_2{\mathbb{R}}$$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties. 

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