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1. Generalized versality, special points, and resolvent degree for the sporadic groups

   https://doi.org/10.1016/j.jalgebra.2024.02.025  

   Gómez-Gonzáles, Claudio; Sutherland, Alexander J; Wolfson, Jesse (June 2024, Journal of Algebra)

   Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cyclic group; an alternating group; a simple factor of a Weyl group of type E6, E7, or E8; or PSL(2,F7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RDk≤d-versality, which we connect to the existence of “special points” on varieties. 

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2. 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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3. Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

   https://doi.org/10.1093/imrn/rnad060  

   Brannan, Michael; Elzinga, Floris; Harris, Samuel J; Yamashita, Makoto (March 2023, International Mathematics Research Notices)

   Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups. 

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4. Weights in a Benson-Solomon block

   https://doi.org/10.1017/fms.2023.53  

   Lynd, Justin; Semeraro, Jason (January 2023, Forum of Mathematics, Sigma)

   Abstract To each pair consisting of a saturated fusion system over a p -group together with a compatible family of Külshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair arises from a genuine block of a finite group algebra in characteristic p , the number of conjugacy classes of weights is supposed to be the number of simple modules in the block. We show that there is unique such pair associated with each Benson-Solomon exotic fusion system, and that the number of weights in a hypothetical Benson-Solomon block is $12$ , independently of the field of definition. This is carried out in part by listing explicitly up to conjugacy all centric radical subgroups and their outer automorphism groups in these systems. 

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5. Bitangents to symmetric quartics

   Bethea, Candace; Brazelton, Thomas (October 2024, arXiv pre-print repository)

   NA (Ed.)

   Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group G=Aut(C), we prove the G-orbits of the bitangents are independent of the choice of C, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective. 

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