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Induced Character
In mathematics, an induced character is the character of the representation ''V'' of a finite group ''G'' induced from a representation ''W'' of a subgroup ''H'' ≤ ''G''. More generally, there is also a notion of induction \operatorname(f) of a class function ''f'' on ''H'' given by the formula :\operatorname(f)(s) = \frac \sum_ f(t^ st). If ''f'' is a character of the representation ''W'' of ''H'', then this formula for \operatorname(f) calculates the character of the induced representation ''V'' of ''G''.. Translated from the second French edition by Leonard L. Scott. The basic result on induced characters is Brauer's theorem on induced characters. It states that every irreducible character on ''G'' is a linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups allow many group-theoretic problems to be reduced to problems in linear algebra. In physics, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the autom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Representation
In group theory, the induced representation is a group representation, representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of '','' the operation of forming induced representations is an important tool to construct new representations''.'' Induced representations were initially defined by Ferdinand Georg Frobenius, Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved. Constructions Algebraic Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the Index of a subgroup, index of in and let be a full set of representatives in of the Coset, left cosets in . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induction Of A Class Function
Induction or inductive may refer to: Biology and medicine * Labor induction (of birth) * Induction chemotherapy, in medicine * Enzyme induction and inhibition * General anaesthesia Chemistry * Induction period, slow stage of a reaction * Inductive cleavage, in organic chemistry * Inductive effect, change in electron density * Asymmetric induction, preferring one stereoisomer over another Computing * Grammar induction * Inductive bias * Inductive probability * Inductive programming * Rule induction * Word-sense induction Mathematics * Backward induction in game theory and economics * Induced representation, in representation theory * Mathematical induction, a method of proof ** Strong induction ** Structural induction ** Transfinite induction *** Epsilon-induction * Parabolic induction Philosophy * Inductive reasoning, in logic Physics * Electromagnetic induction * Electrostatic induction * Forced induction, or turbocharging, of an engine Other uses * Ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugation map on ''G''. Such functions play a basic role in representation theory. Characters The character of a linear representation of ''G'' over a field ''K'' is always a class function with values in ''K''. The class functions form the center of the group ring ''K'' 'G'' Here a class function ''f'' is identified with the element \sum_ f(g) g. Inner products The set of class functions of a group with values in a field form a -vector space. If is finite and the characteristic of the field does not divide the order of , then there is an inner product defined on this space defined by \langle \phi , \psi \rangle = \frac \sum_ \phi(g) \overline, where denotes the order of and the overbar denotes conjugation in the field . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer's Theorem On Induced Characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is part of the representation theory of finite groups. Background A precursor to Brauer's induction theorem was Artin's induction theorem, which states that , ''G'', times the trivial character of ''G'' is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of ''G.'' Brauer's theorem removes the factor , ''G'', , but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups. Another result between Artin's induction theorem and Brauer's induction theorem, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field (mathematics), field, with some generalizations given at the end of the article. Definition Let ''V'' be a vector space over the field ''K''. As usual, we call elements of ''V'' ''vector space, vectors'' and call elements of ''K'' ''scalar (mathematics), scalars''. If v1,...,v''n'' are vectors and ''a''1,...,''a''''n'' are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is :a_1 \mathbf v_1 + a_2 \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Group
In algebra, more specifically group theory, a ''p''-elementary group is a direct product of a finite cyclic group of order relatively prime to ''p'' and a ''p''-group. A finite group is an elementary group if it is ''p''-elementary for some prime number ''p''. An elementary group is nilpotent. Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups. More generally, a finite group ''G'' is called a ''p''-hyperelementary if it has the extension :1 \longrightarrow C \longrightarrow G \longrightarrow P \longrightarrow 1 where C is cyclic of order prime to ''p'' and ''P'' is a ''p''-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary. See also * Elementary abelian group In mathematics, specifically in group theory, an elementary abelian group is an abelian group in whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
