Related papers: Unipotent Generators for Arithmetic Groups

We show the existence of a unitriangular basic set for unipotent blocks simple reductive groups of classical type in bad characteristic with some exceptions. Then,we introduce an algorithm to count irreducible unipotent Brauer characters…

We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.

We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group…

We derive generating functions for the ranks of pre-modular categories associated to quantum groups at roots of unity.

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…

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…

We give a presentation by generators and relations of the group $U_4(\mathbb{Z}[1/\sqrt{2},i])$ of unitary $4\times 4$ matrices with entries in the ring $\mathbb{Z}[1/\sqrt{2},i]$. This is motivated by the problem of exact synthesis for the…

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of…

We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition…

Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$…

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by…

For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.

In this paper we present some inequalities for the order, the exponent, and the number of generators of the c-nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c \geq 1$) of a…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

A connected algebraic group in characteristic 0 is uniquely determined by its Lie algebra. In this paper an algorithm is given for constructing an algebraic group in characteristic 0, given its Lie algebra. Using this an algorithm is…

Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic…

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly…

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set…