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Schur–Weyl Duality
Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pioneers of representation theory of Lie groups, Issai Schur, who discovered the phenomenon, and Hermann Weyl, who popularized it in his books on quantum mechanics and classical groups as a way of classifying representations of unitary and general linear groups. Schur–Weyl duality can be proven using the double centralizer theorem. Description Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of symmetry that determine each other. Consider the tensor space : \mathbb^n\otimes\mathbb^n\otimes\cdots\otimes\mathbb^n with ''k'' factors. The symmetric group ''S''''k'' on ''k'' letters acts on this space (on the left) by permuting the factors, : \sigma(v_1\otimes v_2\otimes\cdots\otimes v_k) = v_\otimes v_\otimes\cdots\otimes v_. The ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Diagram
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Groups
Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a group of people * Representation, in contract law a pre-contractual statement that may (if untrue) result in liability for misrepresentation * Labor representation, or worker representation, the work of a union representative who represents and defends the interests of fellow labor union members * Legal representation, provided by a barrister, lawyer, or other advocate * Lobbying or interest representation, attempts to influence the actions, policies, or decisions of officials * "No taxation without representation", a 1700s slogan that summarized one of the American colonists' 27 colonial grievances in the Thirteen Colonies, which was one of the major causes of the American Revolution * Permanent representation, a type of diplomatic mission ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Evans Howe
Roger Evans Howe (born May 23, 1945) is the William R. Kenan, Jr. Professor Emeritus of Mathematics at Yale University, and Curtis D. Robert Endowed Chair in Mathematics Education at Texas A&M University. He is known for his contributions to representation theory, in particular for the notion of a reductive dual pair and the Howe correspondence, and his contributions to mathematics education. Biography He attended Ithaca High School, then Harvard University as an undergraduate, becoming a Putnam Fellow in 1964. He obtained his Ph.D. from University of California, Berkeley in 1969. His thesis, titled ''On representations of nilpotent groups'', was written under the supervision of Calvin Moore. Between 1969 and 1974, Howe taught at the State University of New York in Stony Brook before joining the Yale faculty in 1974. His doctoral students include Ju-Lee Kim, Jian-Shu Li, Zeev Rudnick, Eng-Chye Tan, and Chen-Bo Zhu. He moved to Texas A&M University in 2015. He has been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Algebra
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley-Lieb algebra, or the group algebra of the symmetric group. Representations of the partition algebra are built from sets of diagrams and from representations of the symmetric group. Definition Diagrams A partition of 2k elements labelled 1,\bar 1, 2,\bar 2,\dots, k,\bar k is represented as a diagram, with lines connecting elements in the same subset. In the following example, the subset \ gives rise to the lines \bar 1 - \bar 4, \bar 4 -\bar 5, \bar 5 - 6, and could equivalently be represented by the lines \bar 1- 6, \bar 4 - 6, \bar 5 - 6, \bar 1 - \bar 5 (for instance). For n\in \mathbb and k\in \mathbb^*, the partition algebra P_k(n) is defined by a \mathbb-basis made of partitions, and a multiplication given by diagram concatenation. The concatenated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Algebra
In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality. Structure The Brauer algebra \mathfrak_n(\delta) is a \mathbbdelta/math>-algebra depending on the choice of a positive integer n. Here \delta is an indeterminate, but in practice \delta is often specialised to the dimension of the fundamental representation of an orthogonal group O(\delta). The Brauer algebra has the dimension :\dim\mathfrak_n(\delta) = \frac = (2n-1)!! = (2n-1)(2n-3)\cdots 5\cdot 3\cdot 1 Diagrammatic definition A basis of \mathfrak_n(\delta) consists of all pairings on a set of 2n elements X_1, ..., X_n, Y_1, ..., Y_n (that is, all perfect matchings of a complete graph K_n: any two of the 2n elements may be matched to each other, regardless of their symbols). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Representation
In mathematics, and in particular the theory of group representations, the regular representation of a group ''G'' is the linear representation afforded by the group action of ''G'' on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. Finite groups For a finite group ''G'', the left regular representation λ (over a field ''K'') is a linear representation on the ''K''-vector space ''V'' freely generated by the elements of ''G'', i. e. they can be identified with a basis of ''V''. Given ''g'' ∈ ''G'', λ''g'' is the linear map determined by its action on the basis by left translation by ''g'', i.e. :\lambda_:h\mapsto gh,\texth\in G. For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given ''g'' ∈ ''G'', ρ''g'' is the linear map on ''V'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maschke's Theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group ''G'' without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character. Formulations Maschke's theorem addresses the question: when is a gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Representation
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity. Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group ''G'' over a field k is a semisimple module over the group ring ''k'' 'G'' Equivalent characterizations Let ''V'' be a representation of a group ''G''; or more generally, let ''V'' be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Representation
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or medical symptoms a sign of disease. A conventional sign signifies by agreement, as a full stop signifies the end of a sentence; similarly the words and expressions of a language, as well as bodily gestures, can be regarded as signs, expressing particular meanings. The physical objects most commonly referred to as signs (notices, road signs, etc., collectively known as signage) generally inform or instruct using written text, symbols, pictures or a combination of these. The philosophical study of signs and symbols is called semiotics; this includes the study of semiosis, which is the way in which signs (in the semiotic sense) operate. Nature Semiotics, epistemology, logic, and philosophy of language are concerned about the nature of sign ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
