辫群与辛 Steinberg 群

On an action of the braid group B_{2g+2} on the free group F_{2g}

We construct an action of the braid group B_{2g+2} on the free group F_{2g} extending an action of B_4 on F_2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B_{2g+2} into the symplectic modular group Sp_{2g}(Z). In the special case g=2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp_4(Z).

A Probabilistic Approach to Conjugacy Classes in the Finite Symplectic and Orthogonal Groups

Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods one obtains a probabilistic proof of Steinberg's count of unipotent matrices and generalizations of formulas of Rudvalis and Shinoda.

Smallest non-cyclic quotients of braid and mapping class groups

We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary p…

On Parabolic Subgroups of Symplectic Reflection Groups

Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. An…

整数辛群的公寓类

Apartment classes of integral symplectic groups

本文通过替代证明展示了辛群Sp_{2n}(Q)的Steinberg模块由整数公寓类生成。

辫子群与 Artin 群中的成员问题

Membership problems in braid groups and Artin groups

本文依据定义图的禁用诱导子图分类了多种成员问题可判定的 Artin 群，并证明辫子群 $\mathbf{B}_n$ 中这些问题可判定当且仅当 $n\leq3$。

Congruence subgroups of braid groups

In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on 3 strands.

From braids to transverse slices in reductive groups

In 1965, Steinberg's study of conjugacy classes in connected reductive groups led him to construct an affine subspace parametrising regular conjugacy classes, which he noticed is also a cross section for the conjugation action by the unipotent radical of a Borel subgroup on another affine subspace. Recently, generalisations of this slice and its cross section property have been obtained by Sevostyanov in the context of quantum group analogues of W-algebras and by He-Lusztig in the context of Deligne-Lusztig varieties. Such slices are often thought of as group analogues of Slodowy slices. In th…

辫群模其同余子群的商群

Quotients of braid groups by their congruence subgroups

该研究解决了Dan Margalit提出的公开问题，确定了辫群同余子群在整一般线性群中的像，并利用辛同余子群刻画了辫群模同余子群的商群结构。

Pairs of Generators for Matrix Groups. I

Matrix generators for the general and special linear groups, the symplectic groups and the general and special unitary groups over finite fields. For the most part the generators have been obtained by translating Steinberg's generators for groups of Lie type into matrix form.

数环的辛群中的最高次数有理上同调

Top-degree rational cohomology in the symplectic group of a number ring

研究数环的辛群在特定条件下的有理上同调性质，揭示其与主理想环的区别，并通过分析辛Steinberg模块的结构得出结论。

无限族中的一般 Hecke 代数

Generic Hecke algebras in the infinite

本文为满足 Steinberg 定理的非本原晶体学复反射群无限族构造极小类 Coxeter 表示，通过配置空间理解 x-关系并实现辫子定理，从而推广至一般 Hecke 代数，并重新审视仿射 GDAHA 族。

The cohomology of the braid group B_3 and of SL_2(Z) with coefficients in a geometric representation

The purpose of this article is to describe the integral cohomology of the braid group B_3 and SL_2(Z) with local coefficients in a classical geometric representation given by symmetric powers of the natural symplectic representation. These groups have a description in terms of the so called "divided polynomial algebra". The results show a strong relation between torsion part of the computed cohomology and fibrations related to loop spaces of spheres.

通过球面的循环覆盖表示辫群：Zariski 闭包与算术性

Representations of braid groups via cyclic covers of the sphere: Zariski closure and arithmeticity

本文研究了纯辫群通过平面曲线族纤维上同调的幺元在定义于有理数上的线性代数群中的本原表示，给出了其像的 Zariski 闭包为极大以及为算术格的判据，推广了前人结果并回答了 McMullen 的问题。