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Topic: Representation of a finite group

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Group representation

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	Question About Representations of Finite Groups \| The n-Category Café
		A note on splitting fields of representations of finite groups.
		By Brauer’s theorem every representation is isomorphic to a subrepresentation of a direct sum of representations induced from representations of p-elementary subgroups.
		That is a finite field extension which makes the algebra a direct sum of matrix algebras (sometimes called a split algebra).
	golem.ph.utexas.edu /category/2007/07/question_about_representations.html (3982 words)

	NationMaster - Encyclopedia: Finite group (Site not responding. Last check: 2007-10-29)
		Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups.
		Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations.
		Properties of groups Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange.
	www.nationmaster.com /encyclopedia/Finite-group (1404 words)

	20: Group Theory and Generalizations
		Groups acting on vector spaces are subgroups of the matrix groups studied in Linear Algebra.
		Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology.
		Nielsen's theorem: subgroups of free groups are free.
	www.math.niu.edu /~rusin/known-math/index/20-XX.html (2774 words)

	Group Representation (Site not responding. Last check: 2007-10-29)
		Representation theory of the PoincarÃ© group - In mathematics, the representation theory of the double cover of the PoincarÃ© group is an example of the theory for a Lie group, in a case that is neither a compact group nor a semisimple group.
		Group representation - Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces.
		Representation theory of the symmetric group - In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.
	so98.maptohealthandwealth.com /grouprepresentation.html (1619 words)

	Finite Group Theory - Cambridge University Press
		The finite simple groups have been classified and are becoming better understood.
		Finite Group Theory develops the foundations of the theory of finite groups.
		It can serve as a text for a course on finite groups for students already exposed to a first course in algebra.
	www.cambridge.org /aus/catalogue/catalogue.asp?isbn=0521786754 (202 words)

	Lie group Summary
		The Lorentz group and the Poincare group of isometries of spacetime are Lie groups of dimensions 6 and 10 that are used in special relativity.
		The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the standard model, whose dimension corresponds to the 1 photon + 3 vector bosons + 8 gluons of the standard model.
		The group of smooth maps from a manifold to a finite dimensional group is called a gauge group, and is used in quantum field theory and Donaldson theory.
	www.bookrags.com /Lie_group (4005 words)

	Finite group Summary
		Thus 1,2,4,8,16,32 is a finite sequence, as is -1,0,1,-1,0,1, and 1,3,5,7,9,...,99.
		Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups.
		Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations.
	www.bookrags.com /Finite_group (1026 words)

	Springer Online Reference Works
		The operation of constructing an induced representation is the simplest and most important stage in the construction of representations of more complicated groups by starting from representations of simpler groups, and for a wide class of groups a complete description of the irreducible representations can be given in terms of induced representations or their generalizations.
		(the theorem on the composition of induced representations).
		A finite group is called monomial if each of its irreducible representations is induced by a one-dimensional representation of some subgroup.
	eom.springer.de /i/i050730.htm (795 words)

	Group Theory and Physics
		Therefore, the character of the reducible representation is the sum of the characters for the irreducible representations that comprise it.
		The electric dipole moment is a vector, and corresponds to the irreducible representation J = 1 of the rotation group.
		Their representations share many of the properties of the representations of finite groups, but the methods of working with the groups is somewhat different.
	www.du.edu /~jcalvert/phys/groups.htm (5735 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-29)
		We may recast a representation of a group \pi on a k-vector space as a module for the group algebra k\pi.
		For \pi a finite group and g a finite dimensional p-restricted Lie algebra, the algebras k\pi and V(g) are examples of a general class of algebras, the group algebras of finite group schemes.
		One means of study of representations is to use cohomology which introduces algebro-geometric invariants associated to the commutative k algebra of even dimensional cohomology.
	www.math.unl.edu /pi/colloquia/abstract-20021003.txt (181 words)

	Yuriy Drozd - Research interests (Site not responding. Last check: 2007-10-29)
		Representations of groups and algebras, especially classification of indecomposable representations, representation type (finite, tame, wild), etc.
		At the same time, I developed the applications of the idèles technique for study of genera of integral representations, in particular, proved the analogues of the theorems on the distribution of prime ideals for the distribution of maximal submodules in a genus.
		We proved that such a classification is always either of finite type, or tame, or wild (in the same sense as in the theory of finite dimensional algebras) and gave a complete description of curves of finite and tame types, as well as of all vector bundles over such curves.
	bearlair.drozd.org /~yuriy/interests.html (1185 words)

	Seminar "Group theory and topology"
		For every finitely generated group G on approach proposed by A.Olshanskii and the author is applied.
		Finitely generated groups G become geometric objects when endowed with the word metric (depending on the generating set): the distance between a and b from G is the length of the shortest word representing a^{-1}b.
		The existence of a bounded-simple 2-generated group, containing a free non-cyclic subgroup, and the existence of an infinite simple bounded-generated 2-generated group are proven.
	math.vanderbilt.edu /~msapir/altopfall02.html (942 words)

	PlanetMath: character
		be a finite dimensional representation of a group
		is finite, the characters of the irreducible representations of
		Cross-references: inner product, orthonormal, multiplication, pointwise addition, basis, complex numbers, irreducible, finite, class function, conjugate, properties, trace, function, field, scalar, vector space, group, representation, finite dimensional
	planetmath.org /encyclopedia/Character.html (117 words)

	Finite Group Theory - Cambridge University Press
		For students already exposed to a first course in algebra, it serves as a text for a course on finite groups.
		For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field.
		It also provides the specialist in finite group theory with a reference on the foundations of the subject.
	www.cambridge.org /us/catalogue/catalogue.asp?isbn=0521786754 (166 words)

	Dissertation Defense, WMU Graduate College
		Graphs, groups, and surfaces are all subjects of study in topological graph theory, using techniques and principles from the disciplines of graph theory, algebra, and topology.
		A Cayley graph provides a graphical representation of a finite group and a fixed generating set for the group; a Cayley map is a two?cell imbedding into a surface of a Cayley graph such that labeled outward?directed darts occur in the same sequence at each vertex.
		It is an extension of a dihedral group by a cyclic group.
	www.wmich.edu /graduate/dissertation/dis-archive/smith.html (411 words)

	New Topological Structures in Physics
		The original McKay correspondence related finite subgroups of Sl(2) and Dynkin diagrams of type ADE; the latter occur as intersection pairings in the cohomology of crepant resolutions of C2/.
		There are two basic invariants of a finite group: its representation ring and the center of its group algebra.
		The center of a group algebra generalizes to the Chen-Ruan orbifold co-homology of X/G. This version of the correspondence defines a conjectural multiplicative equivalence of Chen-Ruan cohomology with a deformation of the cohomology ring of a crepant resolution, involving Gromov-Witten invariants associated to certain exceptional sets.
	www.math.wisc.edu /~shi/topological_structures/McKay_correspondences_w.htm (650 words)

	Weyl Groups
		Note that the 48-element Double Binary Tetrahedral Group and the 96-element Double Binary Octahedral Group are not in 1-1 correspondence with the 72 root vectors of E6 and the 126 root vectors of E7.
		The Weyl group of SU(N) is the permutation group S_N with N! elements.
		DN generates the Lie Group Spin(2N), which is the double-cover of the group of rotations in the 2N-dimensional vector grade-1 part of Cl(2N).
	www.valdostamuseum.com /hamsmith/Weyl.html (5287 words)

	UWM Math: Algebra: Representation Theory
		The theory of algebras and their representations is generally considered to originate in Hamilton's definition of Hypercomplex numbers and his famous construction of the quaternions.
		Maschke's Theorem states that the representations of the group algebras of finite groups over fields of characteristic zero are semisimple, that is that every representation is a direct sum of simple subrepresentations of which there are only finitely many.
		The structure of finite dimensional algebras with the latter property was completely determined by Wedderburn in 1908.
	www.uwm.edu /Dept/Math/Research/Algebra/reptheory/rt.html (753 words)

	[No title]
		The group $G$ acts on $V$ and hence also on $\F[V]$ via the representation $\rho$ and we denote by $\F[V]^G$ the subalgebra of $G$-invariant polynimials in $\F[V]$\/.
		We also illustrate with the example of the quaternion group that these new, fine, orbit Chern classes are sufficient to generate the ring of invariants in cases where orbit Chern classes fail to do so.
		An interesting group representational problem that arises in this connection is the following.
	www.lehigh.edu /dmd1/public/www-data/h106 (1247 words)

	Summer 2001 REU Participants
		As it is well known, every irreducible representation of a finite group is fully determined by its character, or in other words, the trace of the resulting linear transformation.
		Thus, in order to study representations, we may, equivalently, study their characters, which are functions on the group in question.
		Representation theory involves groups structures that lie at the heart of abstract algebra and mathematics as a whole.
	www.math.utah.edu /reu/2001-02reuprojects.html (1601 words)

	Finite state machines in JavaScript, Part 1: Design a widget
		Finite state machines are most useful in situations where behavior is driven by many different types of events, and the response to a particular event depends on the sequence of previous events.
		The events that drive finite state machines can be external to the computer, originating from a keyboard, mouse, timer, or network activity, or they can be internal to the computer, originating from other parts of the application program, or other applications.
		Finite state machines are event-driven, and need a way to hook themselves to the events of interest in their runtime environment.
	www.ibm.com /developerworks/java/library/wa-finitemach1/index.html?ca=drs- (4448 words)

	Creation of Finite Soluble Groups
		The simplest method of producing a pc-presentation for a group is to use one of the built-in construction functions.
		Construct the abelian group defined by the sequence Q = [n_1,..., n_r] of positive integers as a pc-group.
		A map f from the free group of rank n to G is returned as well.
	www.umich.edu /~gpcc/scs/magma/text378.htm (1566 words)

	Springer Online Reference Works
		In the early 1950s group algebras of infinite groups were studied in the context of integer group algebras in algebraic topology, and for the investigation of the structure of groups.
		The group algebra of an ordered group is imbeddable in a skew-field (the Mal'tsev–von Neumann theorem).
		An example is the concept of the cross product of a group and a ring, which retains many properties of a group algebra.
	eom.springer.de /G/g045220.htm (579 words)

	Frohlich biography
		Fröhlich's thesis was in two parts, the first being completed by September 1949 and published as the paper The representation of a finite group as a group of automorphisms on a finite Abelian group (1950).
		The necessary extension of representation theory was published by the author in a previous paper [The representation of a finite group as a group of automorphisms on a finite Abelian group (1950)].
		As a group of at most class two is the direct product of prime power groups of at most class two it is sufficient to study fields whose degree is a prime power.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Frohlich.html (1960 words)

	RIF-WG: W3C Rule Interchange Format Working Group Charter
		The Working Group is chartered to first establish the extensible core and possibly a set of extensions, and then (in Phase 2) to begin to specify additional extensions based on user requirements.
		The Working Group should ensure the rule language is compatible with the use of SPARQL as a language for query of the dataset, that the extension mechanism is compatible with use of the SPARQL protocol for fetching additional datasets, and should aim for compatibility with SPARQL's use of XML datatypes, functions and operators.
		The XML Schema Working Group provides a way to define XML grammars, which may be useful in defining the rule format syntax, and a model for data types and data structures.
	www.w3.org /2005/rules/wg/charter (0 words)

	About "ATLAS of Finite Group representations" (Site not responding. Last check: 2007-10-29)
		Representations of many finite simple groups and related groups such as covering groups and automorphism groups of simple groups.
		Includes experimental generic group pagemaker and an experimental representation search program for sporadic groups.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /library/view/5139.html (57 words)

	Anne Shepler
		Let W be a finite group generated by unitary reflections and A be the set of reflecting hyperplanes.
		The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence.
		We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra.
	www.hist.unt.edu /~ashepler/papers/abstracts.html (629 words)

	NeverEndingBooks » Archive » anabelian geometry - noncommutative.org (Site not responding. Last check: 2007-10-29)
		Each dessin will be part of a finite family of dessins which form one orbit under the Galois action and one needs to find invarians to see whether two dessins might belong to the same orbit.
		the monodromy group of a dessin : the subgroup of the symmetric group
		representations is given by the modular data associated to rational conformal field theories.
	www.neverendingbooks.org /?p=305 (1184 words)

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