Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur …
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small …
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).
This talk is a short survey of some results from the character theory of finite groups which are used for the study of the abstract structure of groups. In particular, …
This talk is a short survey of some results from the character theory of finite groups which are used for the study of the abstract structure of groups. In particular, some results of the author are discussed. We consider the following themes. 1. Some notation and elementary definitions. 2. Character table of a group. 3. Interactions and D-blocks. 4. Zeroes in the character table. 5. Characterization of groups by active fragments of the character table. 6. Semiproportional characters. 1. Further, G is a finite group and C is the field of all complex numbers. If g ∈ G then CG(g) is the centralizer of g in G, g := {x−1gx | x ∈ G} is the conjugacy class of G containing g, and k(G) is the number of conjugacy classes of G. We remember concepts: a representation of G over a field F ; the degree of a representation; the character of a representation; reducible and irreducible representations; the kernel of a representation. The writing D ⊆ G denotes that D is a normal subset of G (i. e. the union of some conjugacy classes of G). Majority of necessary to us concepts and results may be find in [1–3]. 2. A character (irreducible character) of G is a character of some representation (respectively, irreducible representation) of G over C. Irr(G) denotes the set of all irreducible characters of G. Then |Irr(G)| = k(G). If Irr(G) = {χ1, χ2, . . . , χk} and Cl(G) = {g1, g2, . . . , gk}, where k = k(G), then (k × k-matrix) X(G) = (χi(gj)) (k × k-matrix) is a character table of G (X is the Greek Chi). The orthogonality relations in X(G) are significant. Problem 1. To investigate the interdependency of the properties of the character table of a group and the abstract structure of this group. 2a. From G to X(G): For any given group G may be constructed X(G) (see [2, theorem 10]).
Two conjectures proposed (old and somewhat new) by the author elsewhere are discussed in this article.One is concerned with a modular version of the regular character of a finite group …
Two conjectures proposed (old and somewhat new) by the author elsewhere are discussed in this article.One is concerned with a modular version of the regular character of a finite group G, and the second one is concerned with the ratio of the product of the sizes of all conjugacy classes of G and the product of the degrees of all irreducible characters.
Groups are the mathematical objects formally describing our idea of symmetry. They appear naturally acting on vector spaces as groups of invertible matrices. Group representation theory is the branch of …
Groups are the mathematical objects formally describing our idea of symmetry. They appear naturally acting on vector spaces as groups of invertible matrices. Group representation theory is the branch of mathematics that studies such actions. More specifically, character theory studies the trace maps associated to those actions. A fundamental question in the field is to understand how much information about a finite group $G$ and its local subgroups can be extracted from the knowledge of the character theory of $G$. In this note, I will report on recent advances in this topic.
Basic concepts Characters On arithmetical properties of characters Products of characters Induced characters and representations Projective representations Clifford theory Brauer's induction theorems Faithful representations Existence of normal subgroups On sums …
Basic concepts Characters On arithmetical properties of characters Products of characters Induced characters and representations Projective representations Clifford theory Brauer's induction theorems Faithful representations Existence of normal subgroups On sums of degrees of irreducible characters Groups of relatively small height The Brauer-Suzuki theorem Appendices Notes on the bibliography Bibliography Author index Subject index.
The aim of this paper is to describe simple methods to distribute the ordinary irreducible characters of finite groups into p-blocks and calculate the other character values from part of …
The aim of this paper is to describe simple methods to distribute the ordinary irreducible characters of finite groups into p-blocks and calculate the other character values from part of the values of the ordinary irreducible characters for p-blocks.
Introduction 1. Prerequisites from group theory 2. Group representations and character theory 3. Modular representation theory 4. Group order formulas and structure theorem 5. Permutation representations 6. Concrete character tables …
Introduction 1. Prerequisites from group theory 2. Group representations and character theory 3. Modular representation theory 4. Group order formulas and structure theorem 5. Permutation representations 6. Concrete character tables of matrix groups 7. Methods for constructing finite simple groups 8. Finite simple groups with proper satellites 9. Janko group J1 10. Higman-Sims group HS 11. Harada group Ha 9. Thompson group Th Bibliography List of symbols Index.
In the past 15 years, research on abstract group theory has been extensive in China. Many results have been found. Research on the stucture of finite groups is focused on …
In the past 15 years, research on abstract group theory has been extensive in China. Many results have been found. Research on the stucture of finite groups is focused on three aspects: the first, solvable groups (including nilpotent groups and supersolvable groups); the second, the influence of subgroups, quotient groups and the antomorphism group on a finite group; the third, finite simple groups.
Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich …
Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne–Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish–Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers.
This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. …
This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.
We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this …
We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a module over the twisted Drinfeld double $D^\alpha(G)$. We interpret this twisted Drinfeld double in terms of the inertia groupoid of a categorical group.
We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this …
We study the gerbal representations of a finite group $G$ or, equivalently, module categories over Ostrik's category $Vec_G^\alpha$ for a 3-cocycle $\alpha$. We adapt Bartlett's string diagram formalism to this situation to prove that the categorical character of a gerbal representation is a module over the twisted Drinfeld double $D^\alpha(G)$. We interpret this twisted Drinfeld double in terms of the inertia groupoid of a categorical group.
A finite group all of whose complex character values are rational is called a rational group. In this paper, we classify all rational groups whose character degree graphs are disconnected.
A finite group all of whose complex character values are rational is called a rational group. In this paper, we classify all rational groups whose character degree graphs are disconnected.
We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight …
We determine the non-abelian composition factors of the finite groups with Sylow normalizers of odd order. As a consequence, among others, we prove the McKay conjecture and the Alperin weight conjecture for these groups at these primes.
Let G be a finite group and χ∈Irr(G). The codegree of χ is defined as cod(χ)=|G:ker(χ)|χ(1) and cod(G)={cod(χ) | χ∈Irr(G)} is called the set of codegrees of G. In this …
Let G be a finite group and χ∈Irr(G). The codegree of χ is defined as cod(χ)=|G:ker(χ)|χ(1) and cod(G)={cod(χ) | χ∈Irr(G)} is called the set of codegrees of G. In this paper, we show that the set of codegrees of M11,M12,M22,M23, and PSL(3,3) determines the group up to isomorphism.
We study the representation growth of simple compact Lie groups and of $\operatorname {SL}_n(\mathcal {O})$, where $\mathcal {O}$ is a compact discrete valuation ring, as well as the twist representation …
We study the representation growth of simple compact Lie groups and of $\operatorname {SL}_n(\mathcal {O})$, where $\mathcal {O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\operatorname {GL}_n(\mathcal {O})$. This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots. We then show that the twist zeta function of $\operatorname {GL}_n(\mathcal {O})$ exists and has the same abscissa of convergence as the zeta function of $\operatorname {SL}_n(\mathcal {O})$, provided $n$ does not divide $\operatorname {char}{\mathcal {O}}$. We compute the twist zeta function of $\operatorname {GL}_2(\mathcal {O})$ when the residue characteristic $p$ of $\mathcal {O}$ is odd and approximate the zeta function when $p=2$ to deduce that the abscissa is $1$. Finally, we construct a large part of the representations of $\operatorname {SL}_2(\mathbb {F}_q[[t]])$, $q$ even, and deduce that its abscissa lies in the interval $[1, 5/2]$.
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using …
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.
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After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of …
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.
Abstract If a group G is π-separable, where π is a set of primes, the set of irreducible characters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">B</m:mi> <m:mi>π</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> …
Abstract If a group G is π-separable, where π is a set of primes, the set of irreducible characters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">B</m:mi> <m:mi>π</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∪</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">B</m:mi> <m:msup> <m:mi>π</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\operatorname{B}_{\pi}(G)\cup\operatorname{B}_{\pi^{\prime}}(G)} can be defined. In this paper, we prove variants of some classical theorems in character theory, namely the theorem of Ito–Michler and Thompson’s theorem on character degrees, involving irreducible characters in the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">B</m:mi> <m:mi>π</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∪</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">B</m:mi> <m:msup> <m:mi>π</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\operatorname{B}_{\pi}(G)\cup\operatorname{B}_{\pi^{\prime}}(G)} .
In this paper, we introduce the concept of weakly [Formula: see text]-hypercyclically embedded subgroups, and investigate the influence of [Formula: see text]-supplemented subgroups on the structure of chief factors of …
In this paper, we introduce the concept of weakly [Formula: see text]-hypercyclically embedded subgroups, and investigate the influence of [Formula: see text]-supplemented subgroups on the structure of chief factors of finite groups. First, we find a connection between [Formula: see text]-supplemented subgroups and normally embedded subgroups. With the help of this connection, we give some criteria for (weakly) [Formula: see text]-hypercyclically embeddability of normal subgroups of finite groups by using fewer [Formula: see text]-supplemented [Formula: see text]-subgroups with given order. In particular, we not only simplify, but also improve the Main Theorem of [L. Miao and J. Zhang, On a class of non-solvable groups, J. Algebra 496 (2018) 1–10]. Finally, we point out that for a [Formula: see text]-subgroup [Formula: see text] of [Formula: see text], the concept of [Formula: see text]-embedded subgroups coincides with the concept of [Formula: see text]-supplemented subgroups.
A finite group is called a CZ(n)-group if the number of the orders of its quasikernels is n. In this note, we obtain a possible upper bound of derived length …
A finite group is called a CZ(n)-group if the number of the orders of its quasikernels is n. In this note, we obtain a possible upper bound of derived length of the solvable CZ(n)-groups. Also, we give a necessary and sufficient condition for a group to be a CZ(2)-group.
Let \Gamma be a group and r_n(\Gamma) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function \mathcal Z_\Gamma(s) = \sum^\infty_{n=1} r_n(\Gamma)n^{-s} …
Let \Gamma be a group and r_n(\Gamma) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function \mathcal Z_\Gamma(s) = \sum^\infty_{n=1} r_n(\Gamma)n^{-s} . When \Gamma is an arithmetic group satisfying the congruence subgroup property then \mathcal Z_\Gamma(s) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups U of the associated simple group G over the associated local field K . Here we show a surprising dichotomy: if G(K) is compact (i.e. G anisotropic over K ) the abscissa of convergence goes to 0 when \dim G goes to infinity, but for isotropic groups it is bounded away from 0 . As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.
Abstract This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups …
Abstract This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$ -set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type $\mathrm {D}$ and $^2\mathrm {D}$ , a crucial property is the so-called $A'(\infty )$ condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in $\mathrm {Irr}(G)$ . This is part of the stronger $A(\infty )$ condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition $A(\infty )$ for groups of type $\mathrm {D}$ would still satisfy $A'(\infty )$ . This will be used in a second paper to fully establish $A(\infty )$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of $G=\mathrm {D}_{ l,\mathrm {sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.
In this paper we classify the finite solvable groups in which distinct nonlinear monomial characters have distinct degrees.
In this paper we classify the finite solvable groups in which distinct nonlinear monomial characters have distinct degrees.
Let G be a finite group. We prove the solvability of G, by assuming that the average degree of non-monomial irreducible characters in Irr(G) is less than 5/2. The bound …
Let G be a finite group. We prove the solvability of G, by assuming that the average degree of non-monomial irreducible characters in Irr(G) is less than 5/2. The bound is sharp.
The present paper concerns certain divisibility properties of the (integer) values of some polynomials naturally defined from characters and modules of finite groups. A generalization of a theorem of K. …
The present paper concerns certain divisibility properties of the (integer) values of some polynomials naturally defined from characters and modules of finite groups. A generalization of a theorem of K. Brown is obtained in this context.
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small …
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing …
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.
The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide …
The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.