F_4 型复单李代数的结构常数表及其在域与环上 F_4 型 Chevalley 群换位子计算中的应用

Generation of relative commutator subgroups in Chevalley groups. II

In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has n…

The generic character table of a Sylow $p$-subgroup of a finite Chevalley group of type $D_4$

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 $U$.

Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with Applications to Derangements

We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (t…

具有$\mathsf{BC}_\ell$-交换子关系的群

Groups with $\mathsf{BC}_\ell$-commutator relations

本文研究具有$\mathsf{BC}_\ell$根子群的群，证明此类群与奇单位Steinberg群存在同态，且在根子群上诱导同构。

Characters, Commutators and Centers of Sylow Subgroups

The character table of a finite group G determines whether |P:P'|=p^2 and whether |P:Z(P)|=p^2, where P is a Sylow p-subgroup of G. To prove the latter, we give a detailed classification of those groups in terms of the generalized Fitting subgroup.

The commutators of classical groups

In his seminal paper, half a century ago, Hyman Bass established a commutator formula in the setting of (stable) general linear group which was the key step in defining the K_1 group. Namely, he proved that for an associative ring A with identity, E(A)=[E(A),E(A)]=[GL(A),GL(A)] where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings, and for a range of classical and algebraic like-groups, mostly in relation to subnormal subgroups of these groups. The major classical theorems …

根分次群

Root Graded Groups

本文提出根分次群概念以推广蒂茨RGD-系统，证明秩至少为3的根分次群可由满足谢瓦莱换位子公式变体的代数结构坐标化，并借助新提出的蓝图技术实现无特征依赖的证明。

Flatness of the commutator map over SL_n

This paper contributes to the study of the fibers of the commutator map on special linear groups in characteristic zero. Specifically, we show that the fibers over non-central elements all have the same dimension. Also we explain that the fibers over central elements can be of larger dimension and compute how large. We use the character tables of finite general linear groups constructed by J.A. Green to count solutions to the commutator equation $[x,y]=g$ over finite fields and use algebraic geometry to go from characteristic $p$ to characteristic $0$. To deal with fibers over central elements…

Commutators and Cartan subalgebras in Lie algebras of compact semisimple Lie groups

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using the same method, we give a new proof of the following theorem (thus avoiding the classification tables of fundamental weights): in compact semisimple Lie algebras, orthogonal Cartan subalgebras always exist (where one of them can be chosen arbitrarily). Some of the consequences of this theorem are the following. $(i)$ If $L=\operatorname{Lie}(G)$ is such a Li…

New Uniform Diameter Bounds in Pro-$p$ Groups

We give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-$p$ domain (with exponent depending on the dimension); Chevalley groups over a pro-$p$ domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some cons…

Character tables and defect groups

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D'|=4 where D' denotes the commutator subgroup of D. We also propose a new cha…

投射Steinberg pro群

Cosheaves of Steinberg pro-groups

本文研究了Steinberg pro群在Zariski拓扑下的局部性质，证明了与一般线性群、奇单位群和Chevalley群相关的Steinberg pro群满足投射余sheaf性质，并推导了相对Steinberg群的交换子公式。

根分级群再探

Root graded groups revisited

本文研究根分级群的代数结构，描述了不可约根系中rank≥3（排除H3和H4）的变种，并提供其构造方法。

Construction of Fischer's sporadic group Fi'_{24} inside GL_{8671}(13)

In this article we construct an irreducible simple subgroup G = <q, y, t, w> of GL_{8671}(13) from an irreducible subgroup T of GL_{11}(2) isomorphic to Mathieu's simple group M_{24} by means of Algorithm 2.5 of [13]. We also use the first author's similar construction of Fischer's sporadic simple group G_1 = Fi_{23} described in [11]. He starts from an irreducible subgroup T_1 of GL_{11}(2) contained in T which is isomorphic to M_{23}. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer's original 3-transposition group Fi_{24} [6]. It is used here to show that G is isomo…