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Topic: Unitary representation

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Group representation

List of harmonic analysis and representation theory topics

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	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).
		(the theorem on the restriction of an induced representation to a subgroup).
	eom.springer.de /i/i050730.htm (795 words)

	Unitary - Wikipedia, the free encyclopedia
		In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept of the Trinity.
		Unitary state and Unitary authority - types of political regions.
		Note:Some authors use the word "unitary" synonymously with unital, which for a module implies the underlying ring has multiplicative identity which acts as the identity on it, and for an algebra implies that the algebra itself has a mutliplicative identity.
	en.wikipedia.org /wiki/Unitary (167 words)

	Unitary representation - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-10-13)
		In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G.
		One of the pioneers in constructing a general theory of unitary representations was George Mackey.
		A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace.
	en.wikipedia.org.cob-web.org:8888 /wiki/Unitary_representation (480 words)

	Pablo Ramacher
		In [2], the regular representation of a reductive prehomogeneous vector space V was studied, and, in particular, the microlocal structure of certain integral operators associated with that representation was examined.
		The regular representation of a real linear group G acting on a smooth real affine variety M was studied in [1] by examining a certain (g,K)-submodule in the Banach space of continuous functions on M vanishing at infinity.
		It would be a major goal to realize this decomposition for certain unitary representations, and to determine the corresponding intertwining operators, by developing a spectral theory for G-invariant differential operators on M, possibly via appropiate embeddings of eigenfunctions, and the study of Eisenstein series.
	www.uni-math.gwdg.de /ramacher/description.html (787 words)

	[No title]
		By a homotopy representation of S we mean a map from BS into the p-completion of the classi- fying space of the unitary group BU(m)^p.
		3.Homotopy representations of SU(2)n and SU(2)n={ 1} 2-Stubborn subgroups of Ln.
		A representation ' of K is J -invariant iff a total multiplici* ty of irreducible factors of ' in IR+(K) is equal to the total multiplicity of fac *tors in IR-(K).
	hopf.math.purdue.edu /Ziemianski/DI4Rep.txt (6002 words)

	Not Even Wrong » Blog Archive » Some History
		The close connection between the basic ideas of representation theory and of quantum mechanics was quite clear to him, so, unlike Einstein, he enthusiastically adopted the new point of view of quantum physics.
		The theory of continuous groups and their unitary representations ought to be taught in undergraduate physics courses.
		Since all unitary quantum irreps of the diffeomorphism group are anomalous, apart from the trivial one, all interesting GCQTs carry anomalous reps of the diffeomorphism group.
	www.math.columbia.edu /~woit/wordpress/?p=85 (1651 words)

	[No title]
		Understanding the unitary representations of a group can therefore be a powerful ﬁrst step toward solving a variety of analytic and geometric problems.
		One possibility is the transfer of representations from ﬂag domains to cycle spaces by means of double ﬁbration transforms.
		We are particularly motivated by the interaction of the theory of admissible representations of semisimple Lie groups and the theory of homogeneous vector bundles and differential operators.
	www.math.tifr.res.in /~rtrrg06/titles.html (2107 words)

	Group Representation -- from Wolfram MathWorld
		Representations have applications to many branches of mathematics, aside from applications to physics and chemistry.
		Also, special kinds of representations may require that a vector space structure is preserved.
		For instance, a unitary representation is a group homomorphism
	mathworld.wolfram.com /GroupRepresentation.html (182 words)

	Abstract for Lecture II:
		Statistics of anyons are described by the representations of the braid groups.
		One of the most interesting representations of the braid groups is Jones’s unitary representation arising from subfactor theory in the theory of operator algebras.
		The discovery of this representation led Jones to his celebrated polynomial invariant of knots and to the ongoing quantum revolution in topology.
	www.math.ucla.edu /dls/2006/WangL2abs.html (234 words)

	[No title]
		No, such a group would be residually finite: As it is finiteily generated the matrix coefficients of its elements will generate a finitely generated ring so that the group would be a subgroup of GL_2(R), where R is a finitely generated ring.
		For example, if a fg group has a faithful representation by matrices then it has a solvable word problem, it is residually finite, it is virtually torsion-free.
		In fact the existence of a low-dimensional faithful representation is a strong condition.
	www.math.niu.edu /~rusin/known-math/99/findim_rep (804 words)

	CiteULike: Quantum symmetries and the Weyl–Wigner product of group representations (Site not responding. Last check: 2007-10-13)
		In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem.
		However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized.
		However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl\–Wigner product and can be factorized.
	www.citeulike.org /user/arichar6/article/714951 (309 words)

	Rogawski, J.D.: Automorphic Representation of Unitary Groups in Three Variables. (AM-123).
		F.A.Q. Automorphic Representation of Unitary Groups in Three Variables.
		The purpose of this book is to develop the stable trace formula for unitary groups in three variables.
		The stable trace formula is then applied to obtain a classification of automorphic representations.
	pup.princeton.edu /titles/4733.html (168 words)

	Publisher description for Library of Congress control number 94048602 (Site not responding. Last check: 2007-10-13)
		George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group.
		Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups.
		Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.
	www.loc.gov /catdir/description/prin021/94048602.html (252 words)

	Full word coding for information processing (US4500955)
		Words input to an information processing system character-by-character are given unique unitary representations independent of alphanumeric content and order.
		A user selects a particular vocabulary set containing the unique unitary representations of each word in the vocabulary set.
		control logic means for directing such unique unitary representation in place of its corresponding alphanumeric word to a word processing apparatus.
	www.delphion.com /details?pn10=US04500955 (258 words)

	unitary - OneLook Dictionary Search
		Example: "A unitary as opposed to a federal form of government"
		Phrases that include unitary: unitary matrix, unitary system, special unitary matrix, unitary representation, unitary state, more...
		Words similar to unitary: one, unitarily, whole, more...
	www.onelook.com /?w=unitary (271 words)

	UC Davis Math: Prelim Exam Preparation
		Decompose a representation into direct sum of irreducible representations.
		Describe all invariant subspaces of a representation decomposed into a sum of irreps.
		simple group, solvable group, splitting field of a polynomial, field expension, degree of field extension, linear representation, invariant subspace, irreducible and completely reducible representations, character, orthogonal group and orthogonal representation, unitary group and unitary representation, special linear group.
	www.math.ucdavis.edu /grad/gpc/exam_prep (653 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		D(g) is unitary forall g in G. Properties: I will use ` to denote "dagger"= transpose and conjugate * D(g)` D(g) = e * D1(g) not unitary, and D2(g) unitary with D1(g)=M^D2(g)M can occur.
		Theorem: given N dim representation, D, of a finite group G of order h, D is equivalent to a unitary representation of G. Proof: Let X = Sum(i=1,h,D(g)`D(g)).
		forall a in G: D`(a)XD(a)=Sum(i=1,h,D`(a)D`(gi)D(gi)D(a))=Sum(i=1,h,D`(gia)D(gia))=X D`S^2D=S^2 let D~=SDS^ => D~`=(SDS^)`=S^`D`S`=S^D`S D~`D~=I => D~ is unitary and D~ is equivalent to D. Theorem: any non-compact group has no finite dime unitary irreps.
	www.cs.cmu.edu /~jcl/classnotes/math/group_theory/unitary.txt (127 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		A representation is called unitary <=> D(g) is unitary forall g in G. Properties:
		D1(g) not unitary, and D2(g) unitary with D1(g)=M^D2(g)M can occur.
		Theorem: given N dim representation, D, of a finite group G of order h, D is equivalent to a unitary representation of G. Proof:
	www.cs.cmu.edu /People/jcl/classnotes/math/group_theory/unitary.html (104 words)

	DC MetaData for: The moment mapping for a unitary representation (Site not responding. Last check: 2007-10-13)
		DC MetaData for: The moment mapping for a unitary representation
		Abstract: For any unitary representation of an arbitrary Lie
		vectors of the representation into the dual of the Lie algebra.
	www.mat.univie.ac.at /~michor/preprint-shadows/moment.html (71 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #4186797 - Unitary representation operators
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		For a journal article, please see the Resource Relation field.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4186797 (87 words)

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