Rational invariants of certain classical groups over finite fields

Classical groups over finite fields, including symplectic groups, unitary groups and orthogonal groups, are finite simple groups of crucial importance. The structure and order of classical groups are the basic problems in group theory. Geometry of classical groups has been found extremely useful in many fields, such as combinatorial mathematics, coding and cryptographic theory. In this paper, a class of quasi-classical groups are introduced, which partially generalize the concepts of symplectic groups and orthogonal groups. We determine the structure and order of quasi-classical groups. As an application, we discuss the relationship between quasi-classical groups and self-dual normal bases over finite fields.

if i < n and if k is any field with more than two elements. As discussed at this conference, no such stability theorem for other classical groups is known. In this paper, we give a prime-toresidue-characteristic homological stability theorem for classical groups over finite fields (Theorem 2), a stability theorem for the integral homology of classical groups over the algebraic closure of a finite field (Theorem 3), and a stability theorem for the integral homology of the special linear groups for finite fields with more than two elements (Theorem 6). We begin with the following well known stability results for the classical Lie groups. The proof is readily obtained by using , S 4n+3 ~ BSPn and the fibrations S 2n+l * BU n BUn+ 1 ~ BSPn+l, S n . BO n * BOn+ 1.

In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear, symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This generalizes previous result of Basu–Rao–Khanna. Then, as an application, we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups. Finally, by using Swan–Weibel’s homotopy trick, we establish graded analogue of the Local-Global Principle for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes the previous result of Bak–Basu–Rao.

Structure Arising from Induction and Jacquet Modules of Representations of Classical p-Adic Groups

Reconstructing rational functions with FireFly

In this paper, we consider rational functions [Formula: see text] with some minor restrictions over the finite field [Formula: see text] where [Formula: see text] for some prime [Formula: see text] and positive integer [Formula: see text]. We establish a sufficient condition for the existence of a pair [Formula: see text] of primitive normal elements in [Formula: see text] over [Formula: see text] Moreover, for [Formula: see text] and rational functions [Formula: see text] with quadratic numerators and denominators, we explicitly find that there are at most [Formula: see text] finite fields [Formula: see text] in which such a pair [Formula: see text] of primitive normal elements may not exist.

The goal of this paper is to study some Poincare series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables.

Translations of the squares in a finite field and an infinite family of 3-designs

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric, and alternating groups in many cases.

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections.

1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational functions F(x) has a unique infinite continued fraction expansion f = a. + 1/(al + 1/(a2 + * * *)), or more briefly f = [a0; a,, a2, * *.* ],

Rational functions over finite fields having continued fraction expansions with linear partial quotients

In this paper, we define B to be a certain subgroup in a classical group over a finite field and use it to construct non-classical BN-pairs. We also show that this construction is different from the classical one. As an application, we give transcendence bases for the fields of rational invariants of their subgroups B and N.

Generating sets for the multiplicative groups of algebras over finite fields and expander graphs

Constructions of ( q, k, 1) difference families with q a prime power and k = 4,5