Generalized Symmetric Group
In mathematics, the generalized symmetric group is the wreath product S(m,n) := Z_m \wr S_n of the cyclic group of order ''m'' and the symmetric group of order ''n''. Examples * For m=1, the generalized symmetric group is exactly the ordinary symmetric group: S(1,n) = S_n. * For m=2, one can consider the cyclic group of order 2 as positives and negatives (Z_2 \cong \) and identify the generalized symmetric group S(2,n) with the signed symmetric group. Representation theory There is a natural representation of elements of S(m,n) as generalized permutation matrices, where the nonzero entries are ''m''-th roots of unity: Z_m \cong \mu_m. The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of Specht modules; see . Homology The first group homology group – concretely, the abelianization – is Z_m \times Z_2 (for ''m'' odd this is isomorphic to Z_): the Z_m factors (which are ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Specht Module
In mathematics, a Specht module is one of the representations of symmetric groups studied by . They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of the symmetric group on ''n'' points. Definition Fix a partition λ of ''n'' and a commutative ring ''k''. The partition determines a Young diagram with ''n'' boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape λ let \ be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape λ. Consequently, it acts on tabloids, and on the free ''k''-module ''V'' with the tabloids as basis. Given a Young table ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ・ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^ for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element ( singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \op ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a c ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Abelianization
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. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Group Homology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''- simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Roots Of Unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation z^n = 1. Unless otherwise specified, the root ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Wreath Product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H (sometimes known as the ''bottom'' and ''top''), there exist two variants of the wreath product: the unrestricted wreath product A \text H and the restricted wreath product A \text H. The general form, denoted by A \text_ H or A \text_ H respectively, requires that H acts on some set \Omega; when unspecified, usually \Omega = H (a regular wreath product), though a different \Omega is sometimes implied. The two variants coincide when A, H, and \Omega are all finite. Either variant is also denoted as A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode U+2240). The notion gene ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Generalized Permutation Matrices
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is :\begin 0 & 0 & 3 & 0\\ 0 & -7 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt2\end. Structure An invertible matrix ''A'' is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix ''D'' and an (implicitly invertible) permutation matrix ''P'': i.e., :A = DP. Group structure The set of ''n'' × ''n'' generalized permutation matrices with entries in a field ''F'' forms a subgroup of the general linear group GL(''n'', ''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a normal subgroup. I ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |