Clifford Theory Research Articles

Refinements of Dade's Projective Conjecture for p-solvable groups

Dade's Projective Conjecture is known to be true for finite p-solvable groups thanks to work of G.R. Robinson, but remains open in general. Work of Isaacs and Navarro suggested to Uno and Boltje refinements of this conjecture. These refinements were studied for finite p-solvable groups by Glesser. In the present paper, inspired by earlier work of Turull, we propose further refinements of the conjecture that take into account the Schur indices and the elements of the Brauer group. We prove that all these refinements of Dade's Projective Conjecture hold for all finite p-solvable groups. In particular, we obtain that the version of Dade's Projective Conjecture which involves character degree residues modulo p, fields of definition and Schur indices, as well as the full strength of Boltje's Conjecture both hold for all finite p-solvable groups. The proof develops a Clifford theory for normalizers of chains of p-subgroups which allows one to reduce the calculation of the relevant sums to simpler groups.

Endoisomorphisms and character triple isomorphisms

This paper concerns aspects of Clifford Theory of finite groups. In earlier papers, Turull proved that if two finite groups yielded the same element of the Brauer–Clifford group, then there was an endoisomorphism from one group to the other, and furthermore, that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other. The paper adapts the definition of Character Triple Isomorphism so that it involves ordinary and Brauer characters, and it preserves fields of definition, Schur indices, decomposition numbers, and blocks. It is proved that each endoisomorphism yields exactly one character triple isomorphism. Character triple isomorphisms can be composed, restricted, produced by direct sums, extension of fields, and these operations have their parallel for the endoisomorphisms. One goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications.

Clifford Theory for Glider Representations

Classical Clifford theory studies the decomposition of simple $G$-modules into simple $H$-modules for some normal subgroup $H \triangleleft G$. In this paper we deal with chains of normal subgroups $1 \triangleleft G_1 \triangleleft \cdots \triangleleft G_d =G$, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field $K$ and relate different representations of the groups appearing in the chain. Picking some normal subgroup $H \triangleleft G$ one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.

Corrigendum to: “Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup”

At some point, after publication, the author realized that the proof of [3, Theorem 5.2] is incorrect. This proof incorrectly adapts the proof of [1, Theorem 4.8] since [3, (5.5)] is incorrect. Using the same proof outline, we correct the proof of [3, Theorem 5.2].

On Clifford theory with Galois action

Let Gˆ be a finite group, N a normal subgroup of Gˆ and ϑ∈IrrN. Let F be a subfield of the complex numbers and assume that the Galois orbit of ϑ over F is invariant in Gˆ. We show that there is another triple (Gˆ1,N1,ϑ1) of the same form, such that the character theories of Gˆ over ϑ and of Gˆ1 over ϑ1 are essentially “the same” over the fieldF and such that the following holds: Gˆ1 has a cyclic normal subgroup C contained in N1, such that ϑ1=λN1 for some linear character λ of C, and such that N1/C is isomorphic to the (abelian) Galois group of the field extension F(λ)/F(ϑ1). More precisely, having “the same” character theory means that both triples yield the same element of the Brauer–Clifford group BrCliff(G,F(ϑ)) defined by A. Turull.

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adèlic profinite groups of type A 2 . This has consequences for the representation zeta functions of arithmetic groups Γ ⊂ H ( k ) , where k is a number field and H is a k-form of SL 3 : assuming that Γ possesses the strong congruence subgroup property, we obtain precise, uniform estimates for the representation growth of Γ. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL 3 ( o ) and SU 3 ( o ) , where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 × 3 matrices in gl 3 ( o ) and gu 3 ( o ) , where o is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions. Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form SL 3 ( o ) , SU 3 ( o ) , GL 3 ( o ) , and GU 3 ( o ) , arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of ‘Ennola duality’.

Generalized Clifford theory for graded spaces

Assume that Γ is a finite abelian group and ε is an antisymmetric bicharacter on Γ. Let V be a Γ-graded space with a non-degenerate ε-symmetric bilinear form of degree zero. The goal of this paper is to develop a generalized Clifford theory on V. We first introduce the ε-Clifford algebra C(V) and the ε-exterior algebra Λ(V), and then establish an analogue of Chevalley identification between C(V) and Λ(V). Secondly, we extend the non-degenerate bilinear form of degree zero on V to a non-degenerate bilinear form on Λ(V). Finally, as an application, we give a realization of the orthosymplectic ε-Lie algebra.

The Schur–Clifford subgroup of the Brauer–Clifford group

We define a Schur–Clifford subgroup of Turull's Brauer–Clifford group, similar to the Schur subgroup of the Brauer group. The Schur–Clifford subgroup contains exactly the equivalence classes coming from the intended application to Clifford theory of finite groups. We show that the Schur–Clifford subgroup is indeed a subgroup of the Brauer–Clifford group, as are certain naturally defined subsets. We also show that this Schur–Clifford subgroup behaves well with respect to restriction and corestriction maps between Brauer–Clifford groups.

Cyclic Homology and Quantum Orbits

A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.

Clifford theory of characters in induced blocks

We present a new criterion to predict if a character of a finite group extends. Let G be a finite group and p a prime. For N ⊳G, we consider p-blocks b and b of N and NN (D), respectively, with (b ) = b, where D is a defect group of b. Under the assumption that G coincides with a normal subgroup G[b] of G, which was introduced by Dade early 1970’s, we give a character correspondence between the sets of all irreducible constituents of φ and those of (φ′)NG(D) where φ and φ are irreducible Brauer characters in b and b, respectively. This implies a sort of generalization of the theorem of Harris-Knorr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.

On 2 nonsplit extension groups associated with HS and HS:2

The group HS:2 is the full automorphism group of the Higman--Sims group HS. The groups 2^{4.}S_6 and 2^{5.}S_6 are maximal subgroups of HS and HS:2, respectively. The group 2^{4.}S_6 is of order 11520 and 2^{5.}S_6 is of order 23040 and each of them is of index 3 850 in HS and HS:2, respectively. The aim of this paper is to first construct \overline{G} = 2^{5.}S_6 as a group of the form 2^{4.}S_6.2 (that is, \overline{G} = G_1.2) and then compute the character tables of these 2 nonsplit extension groups by using the method of Fischer--Clifford theory. We will show that the projective character tables of the inertia factor groups are not required. The Fischer--Clifford matrices of \overline{G}_1 and \overline{G} are computed. These matrices together with the partial character tables of the inertia factors are used to compute the full character tables of these 2 groups. The fusion of \overline{G}_1 into \overline{G} is also given.

Modular G-graded algebras and G-algebras of endomorphisms

We study Clifford Theory and field extensions for strongly group-graded algebras. In [Turull, A., Clifford theory and endoisomorphisms, J. Algebra 371 (2012), 510–520] and [Turull, A., Endoisomorphisms yield mo-dule and character correspondences, J. Algebra 394 (2013), 7–50] the author introduced the notion of endoisomorphism showing that there is a natural connection between it and Clifford Theory of finite group algebras. An endoisomorphism is an isomorphism between G-algebras of endomorphisms, where G is a finite group. We consider here endoisomorphisms between modules over strongly G-graded algebras. An endoisomorphism induces equivalences of categories with some good compatibility properties (see Theorem ?? and Theorem ?? below).

Clifford theory for association schemes II

The Clifford theorem for finite groups was previously generalized to association schemes. The existence of a bijection was demonstrated but was not constructed. Here we describe the construction of this bijection.

Decomposition numbers for Brauer algebras of type [formula omitted] in characteristic zero

We introduce Brauer algebras associated to complex reflection groups of type G(m,p,n), and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.

Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup

Let G be a finite group with a normal subgroup N that has a p-block b of defect 0. As is well-known, we may assume that b is G-stable. Then we show that there is an extension of G/N by a central cyclic p′-subgroup and a p-block of such that the blocks of G covering b biject with the blocks of that cover and corresponding blocks have several equivalent properties. These results extend the known results in the situation that N is a normal p′-subgroup.

Endoisomorphisms yield module and character correspondences

In an earlier paper (Turull, 2012 [12]) the author introduced the concept of endoisomorphisms and showed its natural connection with Clifford theory of finite groups. Associated with Clifford theory are module and character correspondences. In the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results.

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

We find all irreducible constituents of the Weil representation of a unitary group Um(A) of rank m associated to a ramified quadratic extension A of a finite, commutative, local and principal ring R of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when R is a field.

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of p-adic analytic pro-p groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact p-adic analytic groups, we exhibit a link between the relevant p-adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension. Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups SL3(o), where o is a compact discrete valuation ring of characteristic 0, and of the groups SU3(O,o), where O is an unramified quadratic extension of o. These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type A2. Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type A2 defined over number fields.

Clifford theory and endoisomorphisms

This paper describes the relation between Clifford theory of finite groups over any field, and the Brauer–Clifford group. Let G and G¯ be finite groups, let π:G→G¯ be a surjective group homomorphism with kernel H, and let F be any field. Let K be any extension field of F and let S be an irreducible KH-module. We show that to S is associated in a natural way a specific element [[S,π,F]] of a Brauer–Clifford group defined over π and F. As a tool to prove the existence of this association, and to study its properties, we use endoisomorphisms. These are simply certain isomorphisms of related endomorphism algebras as G¯-algebras over F. We show that two modules for two different finite groups have the same (in an appropriate sense) element of the Brauer–Clifford group if and only if there exists an appropriate endoisomorphism.

Clifford theory for graded fusion categories

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant C e -module categories and extensions of C e -module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of these module categories. We associate a G-crossed product fusion category to each G-invariant C e -module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.