Homological stability for classical groups over finite fields

By a classical group we mean one of the groups G L n ( R ) G{L_n}(R) , G L n ( C ) G{L_n}(C) , G L n ( H ) G{L_n}(H) , U ( p , q ) U(p,\,q) , O n ( C ) {O_n}(C) , O ( p , q ) O(p,\,q) , S O ∗ ( 2 n ) S{O^{\ast }}(2n) , S p 2 n ( C ) S{p_{2n}}(C) , S p 2 n ( R ) S{p_{2n}}(R) , or S p ( p , q ) Sp(p,\,q) . Let G G be a classical group and L L its Lie algebra. For each x ∈ L x \in L we determine the closure of the orbit G ⋅ x G \cdot x (for the adjoint action of G G on L L ). The problem is first reduced to the case when x x is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of G G .

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

Let A be a linear transformation with minimum polynomial mA(t) and characteristic polynomial cA(t). We say that A is cyclic if mA(t) = cA(t), semisimple if mA(t) has no repeated factors, and separable if it is both cyclic and semisimple. The proportions of elements of GL(d, q) which are cyclic or separable have been investigated by, amongst others, Neumann and Praeger [7], Wall [10], and Fulman [4] who also studies semisimple elements. Making use of their results, we gave estimates in [1] for these proportions in SL(d, q). Fulman, Neumann and Praeger [5] have recently found good estimates for the corresponding proportions in the finite classical groups, U(d, q2), Sp(2m, q) and O±(d, q). They raise the question of whether similar methods may be used to calculate the proportions in various subgroups and supergroups of the classical groups. In a series of three papers, we shall show how this may be done. In this, the first paper, we deal with the special unitary groups SU(d, q2), consisting of unitary transformations with determinant 1. In the second paper [2] we shall focus on the conformal groups GSp(2m, q) and GO±(d, q), and in the third paper [3] we shall consider how the method may be extended to a number of groups of orthogonal type, including the special orthogonal group and the derived group of the orthogonal group. Our method in [1] involved the vector space cycle index of Kung [6] and Stong [8], which we describe briefly in Section 2. Similarly, in the current series

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.

Let \( \mathbb{F}_q \) be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over \( \mathbb{F}_q \) and \( \mathbb{F}_q \) be the rational function field over \( \mathbb{F}_q \). We seek to understand the structure of the rational invariant subfield \( \mathbb{F}_q \). In this paper, we prove that \( \mathbb{F}_q \) is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

We study the complex-time Segal-Bargmann transform Bs,τKN on a compact type Lie group KN, where KN is one of the following classical matrix Lie groups: the special orthogonal group SO(N,R), the special unitary group SU(N), or the compact symplectic group Sp(N). Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group U(N). We provide an effective method of computing the action of the Segal-Bargmann transform on trace polynomials, which comprise a subspace of smooth functions on KN extending the polynomial functional calculus. Using these results, we show that as N→∞, the finite-dimensional transform Bs,τKN has a meaningful limit Gs,τ which can be identified as an operator on the space of complex Laurent polynomials.

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.

Effective black-box constructive recognition of classical groups

For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of gin textrm{SL}_n(q) is the product of le k pairwise distinct monic irreducible polynomials over mathbb {F}_q, then every element x of textrm{SL}_n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n=p is prime, if g has order frac{q^p-1}{q-1}, then every non-scalar element x in textrm{SL}_p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

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.

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groups

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are PSL 3(q), q ≠ 4, and PSU 3(q2), q2 ≠ 9, 25. If n = 5 they are PSL 5(q), for all q, and PSU 5(q2), q2 ≥ 9. Also, the soluble group PSU 3(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups.

Factorial [formula omitted]-functions and Tokuyama identities for classical Lie groups

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.