Pairs of Generators for Matrix Groups. I

Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators

The group U_n(F) of all nxn unipotent upper-triangular matrices over F has derived length d := Ceiling(log_2 (n)), equivalently 2^{d-1} < n <= 2^d. We prove that U_n(F) has a 3-generated subgroup of derived length d, and it has a 2-generated subgroup of derived length d if and only if (21/32)* 2^d < n <= 2^d.

The minimal number of generators of a finite semigroup

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semig…

Primitive Curve Lengths on Pairs of Pants

The problem of determining whether or not a non-elementary subgroup of $PSL(2,\CC)$ is discrete is a long standing one. The importance of two generator subgroups comes from Jørgensen's inequality which has as a corollary the fact that a non-elementary subgroup of $PSL(2,\CC)$ is discrete if and only if every non-elementary two generator subgroup is. A solution even in the two-generator $PSL(2,\RR)$ case appears to require an algorithm that relies on a the concept of {\sl trace minimizing} that was initiated by Rosenberger and Purzitsky in the 1970's Their work has lead to many discreteness res…

Weil表示的矩阵生成器

Matrix generators for Weil representations

本文研究了Weil表示的生成矩阵，提供了Sp_{2ℓ}(r)在GL_{r^ℓ}(F)中的显式矩阵生成器，并描述了不可约Weil表示的生成器。

On generators and representations of the sporadic simple groups

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable a in F, M is similar to diag(a Id_h, M_1), where M_1 is cyclic and 0 <= h <= n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.

Generators of the Hecke algebra of $(S_{2n},B_n)$

A set of ring generators for the Hecke algebra of the Gel'fand pair $(S_{2n},B_n)$, where $B_n$ is the hyperoctahedral subgroup of the symmetric group $S_{2n}$ is presented. Various corollaries are given. A conjecture of Sho Matsumoto is proven.