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1. Totally Symmetric Sets

   Caplinger, Noah; Margalit, Dan (January 2025, American Mathematical Society)

   We survey the theory of totally symmetric sets, with applications to homo- morphisms of symmetric groups, braid groups, linear groups, and mapping class groups. 

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2. 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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3. Symmetries of Fano varieties

   https://doi.org/10.1515/crelle-2024-0077  

   Esser, Louis; Ji, Lena; Moraga, Joaquín (October 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension.This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛.We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group.We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety.For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds.Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family.Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities. 

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4. List-Decoding Homomorphism Codes with Arbitrary Codomains

   Babai, Laszlo; Black, Timothy; Wuu, Angela (August 2018, Leibniz international proceedings in informatics)

   The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case. The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}. The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018). We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list. 

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5. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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