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1. Representations and Tensor Product Growth

   https://doi.org/10.1093/imrn/rnac172  

   Larsen, Michael; Shalev, Aner; Tiep, Pham Huu (July 2022, International Mathematics Research Notices)

   Abstract The deep theory of approximate subgroups establishes three-step product growth for subsets of finite simple groups $$G$$ of Lie type of bounded rank. In this paper, we obtain two-step growth results for representations of such groups $$G$$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $$G$$ be a finite simple group of Lie type and $$\chi $$ a character of $$G$$. Let $$|\chi |$$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $$G$$ that are constituents of $$\chi $$. We show that for all $$\delta>0$$, there exists $$\epsilon>0$$, independent of $$G$$, such that if $$\chi $$ is an irreducible character of $$G$$ satisfying $$|\chi | \le |G|^{1-\delta }$$, then $$|\chi ^2| \ge |\chi |^{1+\epsilon }$$. We also obtain results for reducible characters and establish faster growth in the case where $$|\chi | \le |G|^{\delta }$$. In another direction, we explore covering phenomena, namely situations where every irreducible character of $$G$$ occurs as a constituent of certain products of characters. For example, we prove that if $$|\chi _1| \cdots |\chi _m|$$ is a high enough power of $|G|$, then every irreducible character of $$G$$ appears in $$\chi _1\cdots \chi _m$$. Finally, we obtain growth results for compact semisimple Lie groups. 

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2. Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $$2$$-subgroup conjecture

   https://doi.org/10.1090/tran/7590  

   Schaeffer Fry, A. A. (July 2019, Transactions of the American Mathematical Society)

   Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of G. In a previous article, Schaeffer Fry has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions. 

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3. Centers of Sylow subgroups and automorphisms

   https://doi.org/10.1007/s11856-020-2064-2  

   Glauberman, George; Guralnick, Robert; Lynd, Justin; Navarro, Gabriel (September 2020, Israel Journal of Mathematics)

   Suppose that p is an odd prime and G is a finite group having no normal non-trivial p′-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. 

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4. Rigid automorphisms of linking systems

   https://doi.org/10.1017/fms.2021.17  

   Glauberman, George; Lynd, Justin (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $$2$$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G , then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems. 

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5. On Real and Rational Characters in Blocks

   https://doi.org/10.1093/imrn/rnx170  

   Navarro, Gabriel; Robinson, Geoffrey R; Tiep, Pham Huu (August 2017, International Mathematics Research Notices)

   Abstract The principal $$p$$-block of a finite group $$G$$ contains only one real-valued irreducible ordinary character exactly when $$G/{{\bf O}_{p'}(G)}$$ has odd order. For $$p \ne 3$$, the same happens with rational-valued characters. We also prove an analogue for $$p$$-Brauer characters with $$p \geq 3$$. 

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