Character theory, the Glossary

Table of Contents

1. 86 relations: Abelian group, Absolute value, Alfred H. Clifford, Algebraic integer, Algebraically closed field, Association scheme, Basis (linear algebra), Burnside's theorem, Centralizer and normalizer, Character group, Characteristic (algebra), Class function, Classification of finite simple groups, Clifford theory, Commutator subgroup, Complex conjugate, Complex number, Conjugacy class, Conjugate transpose, Coset, Cyclic group, Dihedral group, Dimension (vector space), Direct sum, Dirichlet character, Disjoint union, Emil Artin, Everett C. Dade, Exterior algebra, Feit–Thompson theorem, Ferdinand Georg Frobenius, Field (mathematics), Finite group, Fourier analysis, Frobenius formula, Frobenius reciprocity, Function (mathematics), George Mackey, Graded vector space, Graham Higman, Group (mathematics), Group isomorphism, Group representation, Group theory, Identity element, If and only if, Induced representation, Inner product space, Integer, Irreducible representation, ... Expand index (36 more) »

2. Representation theory of groups

Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

See Character theory and Abelian group

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

See Character theory and Absolute value

Alfred H. Clifford

Alfred Hoblitzelle Clifford (July 11, 1908 – December 27, 1992) was an American mathematician born in St.

See Character theory and Alfred H. Clifford

Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers.

See Character theory and Algebraic integer

Algebraically closed field

In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in) has a root in.

See Character theory and Algebraically closed field

Association scheme

The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance.

See Character theory and Association scheme

Basis (linear algebra)

In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.

See Character theory and Basis (linear algebra)

Burnside's theorem

In mathematics, Burnside's theorem in group theory states that if G is a finite group of order p^a q^b where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.

See Character theory and Burnside's theorem

Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set \operatorname_G(S) of elements of G that commute with every element of S, or equivalently, such that conjugation by g leaves each element of S fixed.

See Character theory and Centralizer and normalizer

Character group

In mathematics, a character group is the group of representations of a group by complex-valued functions. Character theory and character group are representation theory of groups.

See Character theory and Character group

Characteristic (algebra)

In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity.

See Character theory and Characteristic (algebra)

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.

See Character theory and Class function

Classification of finite simple groups

In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic.

See Character theory and Classification of finite simple groups

Clifford theory

In mathematics, Clifford theory, introduced by, describes the relation between representations of a group and those of a normal subgroup.

See Character theory and Clifford theory

Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

See Character theory and Commutator subgroup

Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

See Character theory and Complex conjugate

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Character theory and Complex number

Conjugacy class

In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b.

See Character theory and Conjugacy class

Conjugate transpose

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b).

See Character theory and Conjugate transpose

Coset

In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets.

See Character theory and Coset

Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

See Character theory and Cyclic group

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

See Character theory and Dihedral group

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field.

See Character theory and Dimension (vector space)

Direct sum

The direct sum is an operation between structures in abstract algebra, a branch of mathematics.

See Character theory and Direct sum

Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b.

See Character theory and Dirichlet character

Disjoint union

In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come.

See Character theory and Disjoint union

Emil Artin

Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.

See Character theory and Emil Artin

Everett C. Dade

Everett Clarence Dade is a mathematician at University of Illinois at Urbana–Champaign working on finite groups and representation theory, who introduced the Dade isometry and Dade's conjecture.

See Character theory and Everett C. Dade

Exterior algebra

In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v.

See Character theory and Exterior algebra

Feit–Thompson theorem

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable.

See Character theory and Feit–Thompson theorem

Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

See Character theory and Ferdinand Georg Frobenius

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

See Character theory and Field (mathematics)

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite.

See Character theory and Finite group

Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

See Character theory and Fourier analysis

Frobenius formula

In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn.

See Character theory and Frobenius formula

Frobenius reciprocity

In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting.

See Character theory and Frobenius reciprocity

Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

See Character theory and Function (mathematics)

George Mackey

George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.

See Character theory and George Mackey

Graded vector space

In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.

See Character theory and Graded vector space

Graham Higman

Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory.

See Character theory and Graham Higman

Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

See Character theory and Group (mathematics)

Group isomorphism

In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations.

See Character theory and Group isomorphism

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. Character theory and group representation are representation theory of groups.

See Character theory and Group representation

Group theory

In abstract algebra, group theory studies the algebraic structures known as groups.

See Character theory and Group theory

Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.

See Character theory and Identity element

If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

See Character theory and If and only if

Induced representation

In group theory, the induced representation is a representation of a group,, which is constructed using a known representation of a subgroup. Character theory and induced representation are representation theory of groups.

See Character theory and Induced representation

Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

See Character theory and Inner product space

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See Character theory and Integer

Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho|_W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations.

See Character theory and Irreducible representation

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

See Character theory and Isomorphism

J-invariant

In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable, is a modular function of weight zero for special linear group defined on the upper half-plane of complex numbers.

See Character theory and J-invariant

Kernel (algebra)

In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1).

See Character theory and Kernel (algebra)

Lie algebra

In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity.

See Character theory and Lie algebra

Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

See Character theory and Lie group

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

See Character theory and Mathematical proof

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Character theory and Mathematics

Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

See Character theory and Matrix (mathematics)

Michio Suzuki (mathematician)

was a Japanese mathematician who studied group theory.

See Character theory and Michio Suzuki (mathematician)

Modular representation theory

Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number.

See Character theory and Modular representation theory

Monster group

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000.

See Character theory and Monster group

Monstrous moonshine

In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group M and modular functions, in particular, the ''j'' function.

See Character theory and Monstrous moonshine

Normal subgroup

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part.

See Character theory and Normal subgroup

Order (group theory)

In mathematics, the order of a finite group is the number of its elements.

See Character theory and Order (group theory)

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

See Character theory and Orthonormal basis

Quaternion group

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication.

See Character theory and Quaternion group

Real element

In group theory, a discipline within modern algebra, an element x of a group G is called a real element of G if it belongs to the same conjugacy class as its inverse x^, that is, if there is a g in G with x^g.

See Character theory and Real element

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

See Character theory and Representation theory

Representation theory of finite groups

The representation theory of groups is a part of mathematics which examines how groups act on given structures. Character theory and representation theory of finite groups are representation theory of groups.

See Character theory and Representation theory of finite groups

Richard Brauer

Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician.

See Character theory and Richard Brauer

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Character theory and Ring (mathematics)

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power.

See Character theory and Root of unity

Semisimple Lie algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

See Character theory and Semisimple Lie algebra

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

See Character theory and Simple group

Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Character theory and Subgroup

Subrepresentation

In representation theory, a subrepresentation of a representation (\pi, V) of a group G is a representation (\pi|_W, W) such that W is a vector subspace of V and \pi|_W(g).

See Character theory and Subrepresentation

Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains.

See Character theory and Sylow theorems

Symmetric algebra

In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains, and is, in some sense, minimal for this property.

See Character theory and Symmetric algebra

Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

See Character theory and Tensor product

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Character theory and Topology

Trace (linear algebra)

In linear algebra, the trace of a square matrix, denoted, is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of.

See Character theory and Trace (linear algebra)

Trivial representation

In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector.

See Character theory and Trivial representation

Up to

Two mathematical objects and are called "equal up to an equivalence relation ".

See Character theory and Up to

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

See Character theory and Vector space

Walter Feit

Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory.

See Character theory and Walter Feit

Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

See Character theory and Weight (representation theory)

See also

Representation theory of groups

• Atlas of Lie groups and representations
• B-admissible representation
• Burnside ring
• Character group
• Character theory
• Commutation theorem for traces
• Complementary series representation
• Complex conjugate representation
• Complex representation
• Corepresentations of unitary and antiunitary groups
• Decomposition matrix
• Dual representation
• Fontaine's period rings
• Frobenius–Schur indicator
• G-module
• Gan–Gross–Prasad conjecture
• Gelfand pair
• Gelfand–Raikov theorem
• Group action
• Group representation
• Group ring
• Induced representation
• K-finite
• Matrix coefficient
• McKay conjecture
• Molien's formula
• Monomial representation
• Multiplicity-one theorem
• P-adic Hodge theory
• Partial group algebra
• Positive-definite function on a group
• Projective representation
• Regular representation
• Representation on coordinate rings
• Representation rigid group
• Representation ring
• Representation theory of Lie groups
• Representation theory of diffeomorphism groups
• Representation theory of finite groups
• Schur orthogonality relations
• Schur–Weyl duality
• Springer correspondence
• Tempered representation

References

[1] https://en.wikipedia.org/wiki/Character_theory

Also known as Character (representation theory), Character of a finite group, Character of a group, Character of a group representation, Character of a representation, Character of a representation of a group, Character value, Degree of a character, Group character, Irreducible character, Ordinary character, Ordinary character theory, Orthogonality relation, Orthogonality relations.

, Isomorphism, J-invariant, Kernel (algebra), Lie algebra, Lie group, Mathematical proof, Mathematics, Matrix (mathematics), Michio Suzuki (mathematician), Modular representation theory, Monster group, Monstrous moonshine, Normal subgroup, Order (group theory), Orthonormal basis, Quaternion group, Real element, Representation theory, Representation theory of finite groups, Richard Brauer, Ring (mathematics), Root of unity, Semisimple Lie algebra, Simple group, Subgroup, Subrepresentation, Sylow theorems, Symmetric algebra, Tensor product, Topology, Trace (linear algebra), Trivial representation, Up to, Vector space, Walter Feit, Weight (representation theory).