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Abstract We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction,https://web.math.princeton.edu/~nmk/kt31_11sept.pdf). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember), realizing Weyl groups of type$$E_6$$ $E_6$and$$E_8$$ $E_8$. 

Abstract The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to$$|G|$$ $\left|G\right|$in a logarithmic sense. 

Abstract We prove that the number of conjugacy classes of a finite groupGconsisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup ofG. This is predicted by the Alperin Weight Conjecture. 

Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ $G/O_{p^{\prime }}\left(G\right)\cong C_{11}^2⋊\text{SL}(2,5)$. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ $2\sqrt{p-1}+1-k_p\left(G\right)$, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 

Abstract Let be a finite group and and be different primes. Assume that is odd and . We prove that if divides the degrees of the nonlinear irreducible ‐modular representations, then has a normal ‐complement. 

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.