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Register a new userThe sparsity of character tables of high rank groups of Lie type
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In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.
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We show that the mod $ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $ell$ admits the structure of a module over the mod $ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $ell$-compact fixed point group depending on the order of $q$ mod $ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$.
The congruence subgroup problem for a finitely generated group $Gamma$ and $Gleq Aut(Gamma)$ asks whether the map $hat{G}to Aut(hat{Gamma})$ is injective, or more generally, what is its kernel $Cleft(G,Gammaright)$? Here $hat{X}$ denotes the profinite completion of $X$. In this paper we investigate $Cleft(IA(Phi_{n}),Phi_{n}right)$, where $Phi_{n}$ is a free metabelian group on $ngeq4$ generators, and $IA(Phi_{n})=ker(Aut(Phi_{n})to GL_{n}(mathbb{Z}))$. We show that in this case $C(IA(Phi_{n}),Phi_{n})$ is abelian, but not trivial, and not even finitely generated. This behavior is very different from what happens for free metabelian group on $n=2,3$ generators, or for finitely generated nilpotent groups.
This paper is a continuation of [GLT], which develops a level theory and establishes strong character bounds for finite simple groups of linear and unitary type in the case that the centralizer of the element has small order compared to $|G|$ in a logarithmic sense. We strengthen the results of [GLT] and extend them to all groups of classical type.
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