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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) = \pi(g), _W. A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations. If (\pi, V) is a representation of ''G'', then there is the trivial subrepresentation: :V^G = \. If f: V \to W is an equivariant map In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ... between two representations, then its kernel is a subrepresentation of V and its image is a subrepresentation of W. References * Representation theory {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrix (mathematics), matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The algebraic objects amenable to such a description include group (mathematics), groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the group representation, representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication. Representation theory is a useful method because it reduces problems in abstract algebra to problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real number ... [...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] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Subspace
Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics and physics * Vector (mathematics and physics) ** Row and column vectors, single row or column matrices ** Vector quantity ** Vector space ** Vector field, a vector for each point Molecular biology * Vector (molecular biology), a DNA molecule used as a vehicle to artificially carry foreign genetic material into another cell ** Cloning vector, a small piece of DNA into which a foreign DNA fragment can be inserted for cloning purposes ** Shuttle vector, a plasmid constructed so that it can propagate in two different host species ** Viral vector, a tool commonly used by molecular biologists to deliver genetic materials into cells Computer science * Vector, a one-dimensional array data structure ** Distance-vector routing protocol, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Induction
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case n = k, ''then'' it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the trut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Map
In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action (mathematics), acted on by the same symmetry group, and when the function commutative property, commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of Invariant (mathematics), invariants, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant. In statistical inference, equivariance under statistical transformations of data is an important property of various estimation methods; see invariant estimator for details. In pure mathematics, equivariance is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
