General Unitary Groups Research Articles

A question of Katz and Tiep on representations of finite general unitary groups

Katz and Tiep proved a characterization of the total Weil character of finite special unitary groups for any integer and any prime power q other than in terms of the character values and irreducible constituents. They asked whether a similar characterization can be done for finite general unitary groups In this article, we prove that such characterization for can be done up to endomorphisms and tensoring by real linear characters.

Equivariant Euler characteristics of unitary buildings

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for any odd $n \geq 3$.

Quantum models based on finite groups

We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical content. Since any linear representation of a finite group can be implemented as a subrepresentation of a permutation representation, quantum-mechanical problems can be formulated in terms of permutation groups. Reproducing quantum behavior in the framework of permutation representations of finite groups makes it possible to clarify the meaning of a number of physical concepts.

Modeling Quantum Behavior in the Framework of Permutation Groups

Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.

Irreducible characters of 3′-degree of finite symmetric, general linear and unitary groups

Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of NG(P), where P is a Sylow 3-subgroup of G. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.

English

Let G be a finite group. Let X 1(G) be the first column of the ordinary character table of G. We will show that if X 1(G) = X1(PGU3(q 2)), then G ≅ PGU3(q 2). As a consequence, we show that the projective general unitary groups PGU3(q 2) are uniquely determined by the structure of their complex group algebras.

Conjugacy classes of small sizes in the linear and unitary groups

Abstract. Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in of size up to and classes in of size up to . We then apply these gap results to obtain some bounds and limits concerning the zeta-type function encoding the conjugacy class sizes of these groups.

Computing nilpotent and unipotent canonical forms: a symmetric approach

AbstractLet k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

DADE'S INVARIANT CONJECTURE FOR THE GENERAL UNITARY GROUP GU4(q2) IN DEFINING CHARACTERISTIC

In this paper, we verify Dade's invariant conjecture for the general unitary group GU 4(q2), q a power of an odd prime p, in the defining characteristic p. Together with the results in [3], this completes the proof of Dade's invariant conjecture for the group GU 4(q2) in the defining characteristic.

The Weil-Steinberg character of finite classical groups

Let GGbe one of the groupsGL⁡(n,q)\operatorname {GL}(n,q)orU(n,q)U(n,q), and let HHdenote the subgroupGL⁡(n−1,q)\operatorname {GL}({n-1},q)orU(n−1,q)U(n-1,q), respectively. We show that the restriction of the Steinberg character of GGto HHequals the product of the Weil character and the Steinberg character of HH.

Fermions in NCG-inspired models

Fermions are introduced by means of the bigraded algebra formalism, designed for the construction of models with spontaneously broken gauge symmetry in the general context of non-commutative geometry (NCG). Representations are derived from the general unitary group SU( n+1) employed as a prototype in a C 2,4 algebra. In the explicitly illustrated n=2 case of a model for the electroweak interaction with minimal fermionic content, result from variant approaches in NGG settings are reproduced. The pattern developed augments these approaches in the construction of more involving higher algebras and other classes of groups or SU( n|1) supergroups.