Finite Special Linear Groups Research Articles

Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

The objective of this series is to study metric geometric properties of (coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley topology and observe connections between large scale structure of metric spaces and group properties of Cayley accumulation points. In Part I, we prove that a disjoint union has property A of Yu if and only if all groups appearing as Cayley accumulation points in the space of marked groups are amenable. As an application, we construct two disjoint unions of finite special linear groups (and unimodular linear groups) with respect to two systems of generators that look similar such that one has property A and the other does not admit (fibered) coarse embeddings into any Banach space with nontrivial type (for instance, any uniformly convex Banach space).

ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose $p$-modular reductions have precisely three composition factors. As a consequence we, in particular, complete the description of the integral forms of the simple $\QQ\mathfrak{S}_n$-module labelled by the hook partition $(n-2,1^2)$. Moreover, we investigate the integral forms of the Steinberg module of finite special linear groups $\PSL_2(q)$ over suitable fields of characteristic 0. In the second part of the paper we explicitly determine the Solomon zeta functions of various families of modules and lattices over group algebra, including Specht modules of symmetric groups labelled by hook partitions and the Steinberg module of $\PSL_2(q)$.

An exotic group as limit of finite special linear groups

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.

Strong involutions in finite special linear groups of odd characteristic

Let t be an involution in GL(n,q) whose fixed point space E+ has dimension k between n/3 and 2n/3. For each g∈GL(n,q) such that ttg has even order, 〈ttg〉 contains a unique involution z(g) which commutes with t. We prove that, with probability at least c/log⁡n (for some c>0), the restriction z(g)|E+ is an involution on E+ with fixed point space of dimension between k/3 and 2k/3. This result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic. We discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups.

THE -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$-generated, that is, which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$-generated.

Weil representations of finite general linear groups and finite special linear groups

Weil representations of finite general linear groups and finite special linear groups

On Jordan decomposition of characters for [formula omitted

As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnaféʼs methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztigʼs twisted induction.

Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums

In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via the Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi- dimensional Kloosterman sums in terms of the frequencies of weights in C(SL(n, q)). In particular, when n = 2, this gives a recursive formula for the power moments of Kloosterman sums. We illustrate our results with some examples.

Unitary Space–Time Group Codes: Diversity Sums From Character Tables

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Diversity sum, which is calculated from the Frobenius norm of the difference of two distinct elements in a signal constellation, is the significant parameter to predict a unitary space–time constellation having good performance in low signal-to-noise ratio (SNR). In this correspondence, we propose a method to compute the diversity sum of a unitary group constellation using a character table. Our proposed analysis is simple, requiring only a lookup of the character table. We illustrate our method for the finite special linear groups <emphasis><formula formulatype="inline"><tex>$SL_{2}$</tex></formula></emphasis>, and compare codes with high diversity sum against fixed point free groups at low SNR. We also introduce the notion of a faithful group constellation, that is, one whose diversity sum is greater than <emphasis><formula><tex>$0$</tex></formula></emphasis>. Faithful group constellations may be obtained from any nontrivial character of a group. We describe the method to do so in this correspondence and illustrate it with the example of the finite projective special linear group <emphasis><formula formulatype="inline"><tex>${\rm PSL}_{2}$</tex></formula></emphasis>. </para>

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SL n ( q ) in the non-defining characteristic ℓ, describe restrictions of irreducible modules from GL n ( q ) to SL n ( q ) , classify complex irreducible characters of SL n ( q ) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SL n ( q ) .

Clifford classes for some overgroups of the special linear groups

In [A. Turull, J. Algebra 235 (2001), 275–314], we calculated the Schur index of each of the irreducible characters of the finite special linear groups. In the present paper, we calculate the Schur index of all the irreducible characters of some overgroups of the special linear groups. The overgroups in question are the special linear groups extended by diagonal automorphisms, and the subgroups of the general linear group that contain the special linear group. To each conjugacy class of irreducible characters of the special linear group in each overgroup is associated a Clifford class. The Clifford class controls all the irreducible characters of the overgroup and intermediate subgroups that are related to the given irreducible by Clifford theory. Knowing only the Clifford class, we can parametrize all the irreducible characters of the intermediate subgroups, and compute, for each parametrized irreducible character, its field of character values, as well as its Schur index over each field. We explicitly compute the Clifford class in each case, and deduce from it the information on the Schur index of all the irreducible characters of the overgroups.

The Irreducible Brauer Characters of the Finite Special Linear Groups in Non-describing Characteristics

The irreducible Brauer characters ofSLn(q) are investigated for primeslnot dividingq. They are described in terms of a set of ordinary characters ofSLn(q) whose reductions modulolare a generating set of the additive group of generalized Brauer characters and the decomposition numbers of this set. These decomposition numbers are described by combinatorial means in terms of the decomposition numbers ofGLn(q). The latter have been investigated in great detail the last 15 years and are known completely forn≤10.

Shintani Descent for Special Linear Groups

In this paper, the Shintani descent of irreducible characters of finite special linear groups is discussed. We parametrize irreducible characters ofSLn(Fq) by making use of (modified) generalized Gelfand–Graev characters. Shintani descent of irreducible characters ofSLn(Fqm) is determined by investigating the Shintani descent of generalized Gelfand–Graev characters. Our results hold for arbitrarypandq.

On the cohomology of the finite special linear groups, I

In this paper we continue the investigation of the cohomology of the finite special linear groups. This was addressed in [2], but here we consider exclusively the case that the underlying field has two elements. We shall complete the proof that the results listed in [2, Table I] hold when 4 = 2. Our approach is much the same as in [2] and we refer to that paper for a more thorough orientation to the problem. However, here we shall not relent when nonzero terms appear in our analysis. This will require a rather detailed investigation of some of the groups and their modules. In addition, we shall construct several nonzero cocycles on Sylow 2-subgroups and decide if they are stable for the action of the Weyl group, i.e., if they are restrictions of cocycles from S&+,(2). In fact, we shall construct (on Sylow 2-subgroups) representatives of most of the “sporadic” nonzero classes. At then end of this paper, we list in Table I many of the explicit cocycles we construct, as well as the explicit cocycles constructed in [2]. In the first part of Section 1 we consider the Hochschild-Serre spectral sequence, which is a central tool for our analysis. We show how to decide when nontrivial terms in this sequence arise from nontrivial 1 and 2 cocycles on a maximal parabolic subgroup, and we show how to construct those cocycles when they exist. Later in Section 1 we analyze the I-cohomology of ,~5’,5,+~(2) acting on the exterior powers of the standard module. In Sections 2 and 3 we address the 1-cohomology and 2-cohomology questions, respectively.

On the cohomology of the finite special linear groups, II

On the cohomology of the finite special linear groups, II

The characters of the finite special linear groups

The characters of the finite special linear groups