Clifford Theory Research Articles

Clifford Theory with Schur Indices in Projective Representation Theory#

For finite groups G, it is shown that central simple G-algebras arise naturally from irreducible modules of twisted group algebras over fields of characteristic 0. This leads to an extension of Turull's Clifford theory with Schur indices that applies in the setting of projective representations of finite groups. This theory has the same properties as in the ordinary representation theory setting, perhaps the most interesting of which is a Schur index preserving bijection between certain sets of irreducible projective characters.

Character tables of parabolic subgroups of Steinberg's triality groups

We compute the conjugacy classes of elements and the character tables of the parabolic subgroups of Steinberg's triality groups D 4 3 ( q ) , where q is a power of an odd prime. Our main tools are Clifford theory applied to the Levi decomposition of the parabolic subgroups and the decomposition of restrictions of the unipotent characters of D 4 3 ( q ) . For many of the calculations we use the CHEVIE package.

Formula omitted]-graded Clifford system in the Hecke algebra of type Bn

We extend the representation theory of the q-analogue of the alternating group to the Hecke algebra of type Bn. We show that the Hecke algebra of type Bn has a Z2-graded Clifford system. Using this result, we introduce a new subalgebra in the Hecke algebra of type Bn. This subalgebra is an extension of the q-analogue of the alternating group. The branching rule for the Hecke algebra of type Bn is given by making use of the generalized Clifford theory.

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.

Calculating Clifford classes for characters containing a linear character once

Given a finite group G , a normal subgroup N of G and an irreducible character χ of G , the Clifford class of χ describes the Clifford theory including Schur indices of χ with respect to the normal subgroup N . Assume there is a subgroup U of N and a linear character θ of U with multiplicity 1 in the restriction of χ to U . We describe a method to calculate the Clifford class of χ in certain cases. This method helps, in particular, in the calculation of the Schur indices of the irreducible characters of the groups contained between the general linear and the special linear groups.

Local analysis of the normalizer problem

For a finite group G, and a commutative ring R, the automorphisms of G inducing an inner automorphism of the group ring RG form a group Aut R ( G). Let Aut int( G)=Aut A ( G), where A is the ring of all algebraic integers in C . It is shown how Clifford theory can be used to analyze Aut int( G). It is proved that Aut int( G)/Inn( G) is an abelian group, and can indeed be any finite abelian group. It is an outstanding question whether Aut Z (G)= Inn(G) if G has an abelian Sylow 2-subgroup. This is shown to be true in some special cases, but also a group G with abelian Sylow subgroups and Aut int( G)≠Inn( G) is given.

Structure and representation theory for the double group of the four-dimensional cubic group

Hypercubic groups in any dimension are defined and their conjugate classifications and representation theories are derived. Double group and spinor representation are introduced. A detailed calculation is carried out on the structures of four-dimensional cubic group O4 and its double group, as well as all inequivalent single-valued representations and spinor representations of O4. All representations are derived adopting Clifford theory of decomposition of induced representations. Based on these results, single-valued and spinor representations of the orientation-preserved subgroup of O4 are calculated.

Clifford Theory for G-Functors

When does a G-functor admit a Clifford theory? In this paper we give a simple axiomatic property which characterizes the existence of such a theory. Satisfaction of this condition in the different contexts leads automatically to the satisfaction of the corresponding theorems of Clifford. We apply it to several examples, some of them already known including the character and Burnside rings.

Clifford Theory for Semisimple G-Groups

In this paper a Clifford theory for semisimple G-groups is developed, as a particular case of an abstract Clifford theory for G-functors.

Clifford Theory for Cyclic Quotient Groups

We describe in detail the Clifford theory for cyclic quotient groups. We pay particular attention to the field of definition of the characters involved, as well as their Schur indices.

A module-theoretic approach to Clifford theory for blocks

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.

Casimir-Operator und ein Satz von Reynolds über Charaktergrade

The following result concerning character degrees is familiar. Let H be a normal subgroup of the finite group G with the identity e. Furthermore let $\theta $ be an irreducible complex character of H, and $\chi $ be an irreducible complex character of G with $\big (\theta ^G,\chi \big )_G \ne 0$ . Then the co-degree $|H|/\theta (e)$ divides the co-degree $|G|/\chi (e)$ . The usual proof rests on Schur's theory of projective representations and on Clifford's theory. The purpose of this paper is to give a more elementary proof based on the so-called Casimir operator.

Blocks of fully graded rings

The writing of this paper was precipitated by a curious observation relating the blocks of a G-ring T, for some finite group G, to the blocks of the fixed subring T of G acting on T. We shall say the an arbitrary ring “has finite block theory” if it is a finite direct sum of indecomposable subrings. The curious observation is that a G-ring T has finite block theory whenever its fixed subring T has finite block theory. This is easily proved in Theorem 2.2 below. Incidently the converse statement can be false, as Example 2.4 below shows. A fully G-graded ring R, as defined in [U] or §4 below, is a G-graded ring whose σ-components form a group under multiplication. The centralizer of the identity component R1 in any such R is naturally a G-ring C. The fixed subring C of G in C is exactly the center Z(R) of R (see [U] or §5 below). This allows us to apply the above observation to C, concluding in Theorem 5.6 that R has finite block theory if and only if C does. We can even obtain a one to one correspondence between blocks of R and Gconjugacy classes of blocks of C, whether or not R has finite block theory (see Theorem 5.8 below). When the fully G-graded ring R has finite block theory, we can reproduce Clifford theory for its blocks, as far as the Fong-Reynolds Theorem (Theorem 8.12 below). Following the observation of Ellers [E] that ordinary Clifford theory works just as well over arbitrary G-invariant subalgebras of group algebras, we develop this theory over an arbitary “G-invariant” subring S of R (see §8 below). One intermediate result in an otherwise straightforward development is surprisingly delicate to prove. It is Theorem 6.6 below, which says that an idempotent e of C which is orthogonal to all its G-conjugates e 6= e must lie in the restriction R[H] of R to its

Modules Graded byG-Sets: Duality and Finiteness Conditions

An efficient technique to uncover the relationship between the representations of a group G and the representations of a subgroup H is to Ž . consider gradings by GrH , the class of the left H-cosets of G. This l w x method of study was initiated by Dade in 3 , where he describes the structure of the simple objects in the category R]gr of G-graded R-modules for a G-graded ring R, as a generalization of the classical Clifford theory. A systematic investigation of modules graded by G-sets was started w x w x by Nastasescu et al. in 9 , and then continued in other papers 1, 8, 5 . Ž . Categorical properties of the category G, A, R ]gr of A-graded left R-modules, where R is a G-graded ring and A a left G-set, were studied first, and later a series of finiteness conditions for the objects of this category were considered. One natural question to ask in this framework is the following. If M is Ž . an object of G, A, R ]gr, where A is a finite G-set, and if N is an Ž R-submodule of M, then which finiteness conditions i.e., ascending chain . condition, descending chain condition, existence of Krull dimension on N

Static Modules and Clifford Theories

The aim of this paper is to establish that Alperin's nonnormal Clifford theory is essentially an equivalence between the categories of divisible and projective modules.

A Constructive Brauer-Witt Theorem for Certain Solvable Groups

AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

Clifford theory on the Burnside ring

Clifford theory on the Burnside ring

On Equivalences between Blocks of Group Algebras: Reduction to the Simple Components

A conjecture of Michel Broué states that ifDis an abelian Sylowp-subgroup of a finite groupG, andH=NG(D), then the principal blocks ofGandHare Rickard equivalent. The structure of groups with abelian Sylowp-subgroups, as determined by P. Fong and M. E. Harris, raises the following question: Assuming that Broué's conjecture holds for the simple components ofG, under what conditions does it hold forGitself? Due to the structure ofG, this problem requires mainly the lifting of Rickard complexes top′-extensions of the simple components and the construction of complexes over wreath products. We give here these reduction steps, which may be regarded as a “Clifford theory” of tilting complexes.

Induction functors and stable Clifford theory for Hopf modules

In this paper, we introduce a revised induction functor to study the Hopf modules and the invariant modules for a Hopf comodule algebra. The simple Hopf modules and the stable Clifford theory are discussed.

H-module endomorphism rings

We discuss the endomorphism ring extension of a module over an H-comodule algebra. A necessary and sufficient condition for such an endomorphism ring extension to be Hopf Galois is obtained. The (weak) H-stability for Hopf modules is introduced to study the endomorphism ring extension of a Hopf module. Moreover, the stable Clifford theory for graded rings and modules is extended to the Hopf case.