Clifford Theory Research Articles

Rationality of twist representation zeta functions of compact 𝑝-adic analytic groups

We prove that for any twist rigid compact p p -adic analytic group G G , its twist representation zeta function is a finite sum of terms n i − s f i ( p − s ) n_{i}^{-s}f_{i}(p^{-s}) , where n i n_{i} are natural numbers and f i ( t ) ∈ Q ( t ) f_{i}(t)\in \mathbb {Q}(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G G is moreover a pro- p p group, we prove that its twist representation zeta function is rational in p − s p^{-s} . To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass.

On Height-Zero Characters in p-Constrained Groups

Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ∈Irr(G) have a positive height, and let there be an anchor group Aϕ. We prove that if the normalizer NG(Aϕ) is p-constrained, then Op´(NG(Aϕ))≠{1G}, where Op´(NG(Aϕ)) is the maximal normal p´ subgroup of NG(Aϕ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.

Vertex-source pairs and the Clifford theory of finite group crossed product algebras

Let G be a finite group and let A be a G-crossed product -algebra where is a commutative neotherian local complete ring such that the field is of characteristic p. We study the Clifford Theory of A over the 1-component A 1 of A with emphasis on the vertex-source pairs of indecomposable A-modules. We present the following application of our results (Section 5): Let be a field of characteristic p that is “big enough”, let so that , let that is covered by b and let T be the G-stabilizer of e. Thus e is a central idempotent of AT . As is known, there is a twisted group algebra C of T over k such that the categories C-mod and Ab-mod are equivalent. Let X be an indecomposable (resp. simple) C-module. Then from a vertex-source pair of X, we give a formula for a vertex-source pair of a corresponding indecomposable (resp. simple) Ab-module .

Skew cellularity of the Hecke algebras of type 𝐺(ℓ,𝑝,𝑛)

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are graded skew cellular algebras. In the special case when p = 2 p = 2 this implies that the Hecke algebras of type G ( ℓ , 2 , n ) G(\ell ,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

Local distinction, quadratic base change and automorphic induction for GL n

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual or conjugate-dual representations. When applied to semi-simple representations of the Weil–Deligne group W F ′ of a non Archimedean local field F, and further translated in terms of representations of GL n (F) via the local Langlands correspondence when F has characteristic zero, it yields various statements concerning the behaviour of different types of distinction under quadratic base change and automorphic induction. When F has residual characteristic different from 2, combining of one of the simple results that we obtain with the tiviality of conjugate-orthogonal root numbers ([8]), we recover without using the LLC a result of Serre on the parity of the Artin conductor of orthogonal representations of W F ′ ([23]). On the other hand we discuss its parity for symplectic representations using the LLC and the Prasad and Takloo-Bighash conjecture.

A criterion for the inductive Alperin weight condition

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5).

McKay Quivers and Lusztig Algebras of Some Finite Groups

We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups mathsf {G} subseteq text {GL}(V) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra widetilde {A}(mathsf {G}) of a finite group mathsf {G} subseteq text {GL}(V), which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.

An Introduction to the Ethics of Belief in the Criminal Justice System

In this article, by adopting a descriptive-analytical method, using library tools, carrying out an interdisciplinary study and taking the general meaning of ethics into consideration, the terms thought and belief, their relationship with the ethics of criminal justice and their application in some pillars, are first explained. Then, with an emphasis on a new reading of William Kingdon Clifford's theory of morality, as well as some moral concepts such as, rational responsibility and self-deception, the intellectual, epistemological foundations and beliefs of some elements of the criminal justice system is critiqued. Furthermore, the development and advancement of his moral logic in the application of this important matter in related issues is addressed.The explanation provided is, we believe that in decision-making bodies, and especially in important professions which deal with legislation and litigation, the holders of these rights are morally responsible to be prudent and criticize a belief based on a justifiable reason, and consequently observe the decision, seek the truth and make the right decision based on the facts. This is because they are morally responsible to think more carefully, find more realistic evidence, and to make more informed decisions and judgments in order to achieve a correct and right belief. Furthermore, they must ensure that there is a logical and rational relationship between their belief and reason. Fuzzy Logic was the criterion and logic employed in order to evaluate the ethics of belief.

On support τ-tilting modules over blocks covering cyclic blocks

Support τ-tilting modules are in bijection with two-term tilting complexes for any finite-dimensional symmetric algebra. This fact motivates us to classify support τ-tilting modules over blocks of finite groups. We classify support τ-tilting modules over particular blocks of finite groups using modular representation theoretical approaches including Clifford theory and Green's indecomposability theorem.

Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

Abstract Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that ( k ⁢ N ) ⁢ c {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a ( G , N , c ) {(G,N,c)} -Brauer pair ( R , f R ) {(R,f_{R})} consists of a p-subgroup R of G and a block f R {f_{R}} of ( k ⁢ C N ⁢ ( R ) ) {(kC_{N}(R))} . If Q is a defect group of c and f Q ∈ 𝐵 ⁢ ℓ ⁡ ( k ⁢ C N ⁢ ( Q ) ) {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))} , then ( Q , f Q ) {(Q,f_{Q})} is a ( G , N , c ) {(G,N,c)} -Brauer pair. The ( G , N , c ) {(G,N,c)} -Brauer pairs form a (finite) poset. Set H = N G ⁢ ( Q , f Q ) {H=N_{G}(Q,f_{Q})} so that ( Q , f Q ) {(Q,f_{Q})} is an ( H , C N ⁢ ( Q ) , f Q ) {(H,C_{N}(Q),f_{Q})} -Brauer pair. We extend Lemma 8.6.4 of the above book to show that if ( U , f U ) {(U,f_{U})} is a maximal ( G , N , c ) {(G,N,c)} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} , then ( U , f U ) {(U,f_{U})} is a maximal ( H , C N ⁢ ( c ) , f Q ) {(H,C_{N}(c),f_{Q})} -Brauer pair containing ( Q , f Q ) {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of ℱ ( U , f U ) ⁢ ( G , N , c ) {\mathcal{F}_{(U,f_{U})}(G,N,c)} and ℱ ( U , f U ) ⁢ ( H , C N ⁢ ( Q ) , f Q ) {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including ( Q , f Q ) {(Q,f_{Q})} and ( U , f U ) {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

On the Alperin-McKay conjecture for simple groups of type A

In this paper characters of the normalizer of d-split Levi subgroups in SLn(q) and SUn(q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ∤6(q2−1) and the Alperin weight conjecture for ℓ-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SLn(q) and SUn(q).

A natural Morita equivalence reduction for some blocks of G-crossed product -lattices and applications to the Clifford theory of finite group modular representation theory

Let be a complete discrete valuation ring with an algebraically closed residue field of characteristic p. Let G be a finite group and let A be a G-crossed product finitely generated -lattice.Section 1 presents a “natural Morita equivalence” reduction for some blocks of A. This setting was pioneered by E. C. Dade and extends some results of A. Hida and S. Koshitani, for example.In Section 2, we apply results in Section 1 and a basic result of Knorr to Finite Group Block Theory. In particular, we obtain a “lift from k to ” of fundamental results of Külshammer. This latter result has recently also been obtained by F. Eisele. We also present an “axiomitization” of the proof of an important result of B. Külshammer and some examples.

The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

Clifford theory of Weil representations of unitary groups

Abstract Let 𝒪 {\mathcal{O}} be an involutive discrete valuation ring with residue field of characteristic not 2. Let A be a quotient of 𝒪 {\mathcal{O}} by a nonzero power of its maximal ideal, and let * {*} be the involution that A inherits from 𝒪 {\mathcal{O}} . We consider various unitary groups 𝒰 m ⁢ ( A ) {\mathcal{U}_{m}(A)} of rank m over A, depending on the nature of * {*} and the equivalence type of the underlying hermitian or skew hermitian form. Each group 𝒰 m ⁢ ( A ) {\mathcal{U}_{m}(A)} gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of 𝒰 m ⁢ ( A ) {\mathcal{U}_{m}(A)} with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.

Supercharacter theories for Sylow p-subgroups of Lie type G 2

We construct a supercharacter theory, and establish the supercharacter table for Sylow p-subgroups of the Chevalley groups of Lie type when p>2. Then we calculate the conjugacy classes, determine the complex irreducible characters by Clifford theory, and obtain the character tables for when p>3.

Clifford theory for Yokonuma-Hecke algebras and deformation of complex reflection groups

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the Yokonuma--Hecke algebras, and in turn with an algebra defined by Aicardi and Juyumaya known as the algebra of braids and ties. The second one can be seen as new deformations of complex reflection groups of type G(d,p,n). We provide several presentations for both algebras and a complete study of their representation theories using Clifford theory.

A VARIANT OF HARISH-CHANDRA FUNCTORS

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

Glider Representations of Group Algebra Filtrations of Nilpotent Groups

We continue the study of glider representations of finite groups $G$ with given structure chain of subgroups $e \subset G_1 \subset \ldots \subset G_d = G$. We give a characterization of irreducible gliders of essential length $e \leq d$ which in the case of $p$-groups allows to prove some results about classical representation theory. The paper also contains an introduction to generalized character theory for glider representations and an extension of the decomposition groups in the Clifford theory. Furthermore, we study irreducible glider representations for finite nilpotent groups.

CLASSIFICATION OF INDECOMPOSABLE MODULES FOR FINITE GROUP SCHEMES OF DOMESTIC REPRESENTATION TYPE

We investigate the representation theory of domestic group schemes $$ \mathcal{G} $$ over an algebraically closed field of characteristic p > 2. We present results about filtrations of induced modules, actions on support varieties, Clifford theory for certain group schemes and applications of Clifford theory for strongly group graded algebras to the structure of Auslander-Reiten quivers. The combination of these results leads to the classification of modules belonging to the principal block of the group algebra k $$ \mathcal{G} $$ .

Clifford theory on rational Cherednik algebras of imprimitive groups

Ram and Rammage have introduced an automorphism and Clifford theory on affine Hecke algebras. Here we will extend them to cyclotomic Hecke algebras and rational Cherednik algebras.