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E8 (mathematics)

Index E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. [1]

120 relations: Adjoint representation, Algebraic group, American Institute of Mathematics, An Exceptionally Simple Theory of Everything, Anomaly (physics), Antony Garrett Lisi, ATLAS of Finite Groups, Atlas of Lie groups and representations, Automorphism, Élie Cartan, Basis (linear algebra), Bulletin of the American Mathematical Society, Cartan matrix, Cartan subalgebra, Cartan subgroup, Chevalley basis, Classification of finite simple groups, Cobalt, Compact group, Compact space, Complex dimension, Conjugacy class, Covering group, David Kazhdan, Deligne–Lusztig theory, Dempwolff group, Determinant, Differential structure, Dimension, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 lattice, E8 manifold, Electron magnetic moment, En (Lie algebra), Euclidean space, F4 (mathematics), Field (mathematics), Finite field, Fokko du Cloux, Freudenthal magic square, Fundamental interaction, Fundamental representation, G2 (mathematics), Galois cohomology, Gauge theory, George Lusztig, Graduate Texts in Mathematics, Group of Lie type, ..., Half-integer, Hans Freudenthal, Heterotic string theory, Inner product space, Integer, Inventiones Mathematicae, Irreducibility (mathematics), Irreducible representation, Isometry group, Jacobi identity, Jacques Tits, Jeffrey Adams (mathematician), John C. Baez, Journal of Algebra, Kazhdan–Lusztig polynomial, Killing form, Lang's theorem, Lattice (group), Lie group, List of finite simple groups, List of simple Lie groups, Mathematics, Mathematische Annalen, Matrix (mathematics), Maximal subgroup, Maximal torus, Michael Freedman, Monster group, New Scientist, Niobium, Notices of the American Mathematical Society, Octonion, On-Line Encyclopedia of Integer Sequences, Orthogonality, Perfect field, Permutation, Real form (Lie theory), Reductive group, Reflection (mathematics), Representation theory, Restricted representation, Root system, Schur multiplier, Science (journal), Scientific American, Semiregular polytope, Simple group, Simple Lie group, Spinor, Spontaneous symmetry breaking, Springer Science+Business Media, Standard Model, String theory, Supergravity, Symmetric space, Symmetry group, Tensor product, Theoretical physics, Thompson sporadic group, Thorold Gosset, Topological manifold, Triviality (mathematics), U-duality, Unimodular lattice, University of Chicago Press, University of Texas at Austin, Weyl character formula, Weyl group, 4 21 polytope, 4-manifold. Expand index (70 more) »

Adjoint representation

In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.

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Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.

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American Institute of Mathematics

The American Institute of Mathematics (AIM) was founded in 1994 by John Fry, co-founder of Fry's Electronics, and located in the Fry's Electronics San Jose, California location.

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An Exceptionally Simple Theory of Everything

"An Exceptionally Simple Theory of Everything" is a physics preprint proposing a basis for a unified field theory, often referred to as "E8 Theory", which attempts to describe all known fundamental interactions in physics and to stand as a possible theory of everything.

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Anomaly (physics)

In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory.

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Antony Garrett Lisi

Antony Garrett Lisi (born January 24, 1968), known as Garrett Lisi, is an American theoretical physicist and adventure sports enthusiast.

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ATLAS of Finite Groups

The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with corrections in 2003.

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Atlas of Lie groups and representations

The Atlas of Lie Groups and Representations is a mathematical project to solve the problem of the unitary dual for real reductive Lie groups.

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In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Élie Cartan

Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.

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Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

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Cartan matrix

In mathematics, the term Cartan matrix has three meanings.

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Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if \in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak).

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Cartan subgroup

In mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebra is a Cartan subalgebra.

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Chevalley basis

In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers.

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Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below.

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Cobalt is a chemical element with symbol Co and atomic number 27.

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Compact group

In mathematics, a compact (topological) group is a topological group whose topology is compact.

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Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

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Complex dimension

In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or a complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d.

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Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.

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Covering group

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p: G → H is a continuous group homomorphism.

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David Kazhdan

David Kazhdan (דוד קשדן) or Každan, Kazhdan, formerly named Dmitry Aleksandrovich Kazhdan (until he left the Soviet Union; Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory.

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Deligne–Lusztig theory

In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by.

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Dempwolff group

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520.

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In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

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In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).

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E6 (mathematics)

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.

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E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

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E8 lattice

In mathematics, the E8 lattice is a special lattice in R8.

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E8 manifold

In mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice.

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Electron magnetic moment

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge.

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En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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F4 (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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Fokko du Cloux

Fokko du Cloux (20 December 1954, Rheden – 10 November 2006) was a Dutch mathematician and computer scientist.

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Freudenthal magic square

In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).

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Fundamental interaction

In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions.

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Fundamental representation

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight.

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G2 (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups.

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Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups.

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Gauge theory

In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.

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George Lusztig

George Lusztig (born Gheorghe Lusztig, May 20, 1946) is a Romanian-American mathematician and Abdun Nur Professor at the Massachusetts Institute of Technology (MIT).

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Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

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Group of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.

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In mathematics, a half-integer is a number of the form where n is an integer.

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Hans Freudenthal

Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician.

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Heterotic string theory

In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string.

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Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

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Irreducibility (mathematics)

In mathematics, the concept of irreducibility is used in several ways.

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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 subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.

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Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.

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Jacobi identity

In mathematics the Jacobi identity is a property of a binary operation which describes how the order of evaluation (the placement of parentheses in a multiple product) affects the result of the operation.

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Jacques Tits

Jacques Tits (born 12 August 1930 in Uccle) is a Belgium-born French mathematician who works on group theory and incidence geometry, and who introduced Tits buildings, the Tits alternative, and the Tits group.

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Jeffrey Adams (mathematician)

Jeffrey David Adams (born 1955) is a mathematician at the University of Maryland who works on unitary representations of reductive Lie groups, and who led the project Atlas of Lie groups and representations that calculated the characters of the representations of E8.

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John C. Baez

John Carlos Baez (born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California.

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Journal of Algebra

Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra.

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Kazhdan–Lusztig polynomial

In the mathematical field of representation theory, a Kazhdan–Lusztig polynomial P_(q) is a member of a family of integral polynomials introduced by.

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Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

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Lang's theorem

In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field \mathbf_q, then, writing \sigma: G \to G, \, x \mapsto x^q for the Frobenius, the morphism of varieties is surjective.

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Lattice (group)

In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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List of finite simple groups

In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.

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List of simple Lie groups

In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Mathematische Annalen

Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Maximal subgroup

In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.

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Maximal torus

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.

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Michael Freedman

Michael Hartley Freedman (born 21 April 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara.

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Monster group

In the area of modern 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 The finite simple groups have been completely classified.

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New Scientist

New Scientist, first published on 22 November 1956, is a weekly, English-language magazine that covers all aspects of science and technology.

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Niobium, formerly known as columbium, is a chemical element with symbol Nb (formerly Cb) and atomic number 41.

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Notices of the American Mathematical Society

Notices of the American Mathematical Society is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue.

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In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

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On-Line Encyclopedia of Integer Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences.

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In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Perfect field

In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.

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In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.

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Real form (Lie theory)

In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers.

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Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field.

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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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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.

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Restricted representation

In mathematics, restriction is a fundamental construction in representation theory of groups.

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Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.

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Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2(G, Z) of a group G. It was introduced by in his work on projective representations.

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Science (journal)

Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.

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Scientific American

Scientific American (informally abbreviated SciAm) is an American popular science magazine.

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Semiregular polytope

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.

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Simple group

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

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Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

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Spontaneous symmetry breaking

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Standard Model

The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.

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String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

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In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity where supersymmetry obeys locality; in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model.

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Symmetric space

In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.

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Symmetry group

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.

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Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

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Theoretical physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena.

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Thompson sporadic group

In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order.

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Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

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Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

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Triviality (mathematics)

In mathematics, the adjective trivial is frequently used for objects (for example, groups or topological spaces) that have a very simple structure.

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In physics, U-duality (short for unified duality) is a symmetry of string theory or M-theory combining S-duality and T-duality transformations.

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Unimodular lattice

In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1.

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University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

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University of Texas at Austin

The University of Texas at Austin (UT, UT Austin, or Texas) is a public research university and the flagship institution of the University of Texas System.

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Weyl character formula

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.

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Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.

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4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.

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In mathematics, a 4-manifold is a 4-dimensional topological manifold.

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E8 (Mathematics), E8 (group), E8 Lie algebra, E8 Lie group, E8 shape, E₈, E₈ (mathematics), Lie group E8.


[1] https://en.wikipedia.org/wiki/E8_(mathematics)

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