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Topic: Representation theory

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	Articles - Group representation (Site not responding. Last check: 2007-11-06)
		Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces.
		Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which is a very well-understood theory.
		Representation theory also depends heavily on the type of vector space on which the group acts.
	www.lastring.com /articles/Group_representation?mySession=803e345e68859f20229fbfe451cc388f (1506 words)

	Representation Theory (Site not responding. Last check: 2007-11-06)
		Representation theory studies the way in which a given group may act on vector spaces; in other words, it is concerned with representing groups as groups of matrices.
		The main questions in representation theory are related to: (1) the construction of irreducible representations; (2) the calculation of certain algebraic invariants of these; (3) the decomposition of other representations into irreducibles.
		Representation theory is a fundamental tool for studying group symmetry - geometric, analytic, or algebraic - by means of linear algebra.
	math.albany.edu:8000 /math/pers/lenart/teach/repres.html (399 words)

	Warwick Mathematics
		Representation theory is the study of representations of algebraic objects by matrices.
		Representation theory research at Warwick centers around objects of Cartan-Dynkin-Weyl type, i.e., reductive algebraic groups, semisimple Lie algebra, finite groups of Lie type, quantum groups, Hecke algebras, Kac-Moody algebras etc. We are particularly interested in representations of affine Hecke algebras, Lusztig conjectures, and combinatorics related to crystal and canonical bases.
		Representation theory is actively used by mathematicians working in many areas, for instance, soluble groups, homological algebra, K-theory, McKay correspondence, string theory, symplectic geometry.
	www.maths.warwick.ac.uk /research/research_areas/rep_thy.html (162 words)

	Harmonic Analysis and Representation Theory (Site not responding. Last check: 2007-11-06)
		Some of the earliest 20th century research in harmonic analysis and representation theory, by the mathematical physicist Eugene Wigner, and by mathematicians such as Marshall Stone and John von Neumann, was actually aimed at better understanding of the mathematical foundations of quantum mechanics.
		Mathematicians working in Representation Theory often concentrate their efforts in a particular family of groups, since each family has its own idiosyncracies and exhibits different properties.
		With its roots deeply embedded in algebra, analysis, and mathematical physics, harmonic analysis and representation theory is an extremely rich subject for investigation, interacting with many parts of both pure and applied mathematics.
	www.math.lsu.edu /grad/harmgrp.html (476 words)

	GROUP REPRESENTATION THEORY FOR PHYSICISTS (Site not responding. Last check: 2007-11-06)
		The unique feature of the approach is that it is based on Dirac's complete set of commuting operators theory in quantum mechanics and thus the representation theories for finite groups, infinite discrete groups and Lie groups are all unified.
		The theory on roots and weights in Lie groups is reformulated in the spirit of representation theory of quantum mechanics.
		The applications of group theory to many-body problem are introduced with emphasis on the various dynamic symmetry models of nuclei.
	www.worldscibooks.com /physics/0262.htm (198 words)

	Psycoloquy 4(11): Representation and the Foundations of Cognitive Science (Site not responding. Last check: 2007-11-06)
		Representation and other intentional concepts just do not seem to be the right sorts of concepts on which to base explanations.
		It would be nice to have a theory of representation that alleviated such worries about the legitimacy of using intentional concepts in explanatory roles.
		A tighter circle is rarely found: an understanding of mental representation presupposes an understanding of cognitive agents, and an understanding of cognitive agents presupposes an understanding of mental representation.
	www.cogsci.ecs.soton.ac.uk /cgi/psyc/newpsy?4.11 (1227 words)

	PlanetMath: representation theory of $\mathfrak{sl}_2 \mathbb{C}$
		is a very important tool for understanding the structure theory and representation theory of other Lie algebras (semi-simple finite dimensional Lie algebras, as well as infinite dimensional Kac-Moody Lie algebras).
		Cross-references: direct sum, completely reducible, Weyl's theorem, simple, operator, eigenvalue, highest weight, dimension, isomorphism, action, vectors, spanned by, integers, bijection, irreducible, infinite dimensional, finite dimensional, semi-simple, structure, theory, representation, relations, satisfy, commutator, Lie bracket, span, Lie algebra
		This is version 4 of representation theory of
	planetmath.org /encyclopedia/RepresentationTheoryOfMathfraksl_2MathbbC.html (208 words)

	IPAM Conformal Field Theory Program
		Methods of quantum field theory may be applied to problems of topological invariants of knots and instanton moduli spaces as well and have led to a wealth of new results in pure mathematics.
		The classification of two-dimensional conformal field theories is thereby reduced to problems in the representation theory of infinite-dimensional Virasoro, Kac-Moody and vertex operator algebras.
		The perturbative expansion of string theory in powers of the coupling constant may be formulated in terms of conformal field theories on families of two-dimensional compact Riemann surfaces, whose genus is the order of the expansion.
	www.ipam.ucla.edu /programs/cft2001 (1082 words)

	University of Toronto Number Theory/Representation Theory Seminar (Site not responding. Last check: 2007-11-06)
		The University of Toronto Number Theory/Representation Theory Seminar
		Recent developments in the theory of motivic integration
		Twisted Characters of Depth Zero Supercuspidal Representations of GL(n).
	www.math.toronto.edu /kc/ntrts (53 words)

	MTH-3E20 : Representation Theory
		Representation Theory demonstrates the enormous power of linear algebra and builds upon what students have learned at level 1 and 2 in Pure Mathematics I and II and Algebra I and II.
		The idea of equivalence of representations is crucial for the theory, although initially quite difficult to grasp.
		The theory of induced representations introduces students to efficient and practical methods of constructing and analysing representations.
	www.mth.uea.ac.uk /maths/syllabuses/0001/3E2001.html (534 words)

	Representation Theory
		The representations returned are inequivalent and consist of all distinct representations, subject to the conditions imposed.
		The representations are found using Schur's method of climbing the composition series for G defined by the pc-presentation.
		The irreducible representation of dimension 4 is not absolutely irreducible, as over GF(4) it splits into two Galois-equivalent representations.
	www.math.niu.edu /help/math/magmahelp/text346.html (789 words)

	Math 961 : Representation Theory Syllabus (Site not responding. Last check: 2007-11-06)
		This course is an introduction to finite-dimensional representations of groups and Lie algebras.
		Representation theory is a flourishing branch of modern mathematics that plays an important role in many recent developments in mathematics and theoretical physics.
		Information about such representations and their invariants is helpful in understanding various group symmetries (think about symmetries of Platonic solids, finite sets, or vector spaces).
	www.math.unh.edu /~nikshych/961-syllabus.html (290 words)

	Fundamentals of Infinite Dimensional Representation Theory
		Infinite dimensional representation theory blossomed in the latter half of the twentieth century, developing in part with quantum mechanics and becoming one of the mainstays of modern mathematics.
		Fundamentals of Infinite Dimensional Representation Theory provides an accessible account of the topics in analytic group representation theory and operator algebras from which much of the subject has evolved.
		Beyond serving as both a general reference and as a text for those requiring a background in group-operator algebra representation theory, for careful readers, this monograph helps reveal not only the subject's utility, but also its inherent beauty.
	www.ramex.com /title.asp?id=6763 (223 words)

	Semantics
		Discourse Representation Theory (DRT) [Kam81,KR93], as the name implies, has taken the notion of an intermediate representation as an indispensable theoretical construct, and, as also implied, sees the main unit of description as being a discourse rather than sentences in isolation.
		This makes it possible for the same representation to be used in applications like translation, which can often be carried out without reference to context, as well as in database query, where the context-dependent elements must be resolved in order to know exactly which query to submit to the database.
		The ability to operate with underspecified representations of this type is essential for computational tractability, since the task of spelling out all of the possible alternative fully specified interpretations for a sentence and then selecting between them would be computationally intensive even if it were always possible in practice.
	cslu.cse.ogi.edu /HLTsurvey/ch3node7.html (2729 words)

	Read This: Briefly Noted, January 2005
		Representation Theory of Finite Reductive Groups is the first book in a new series from Cambridge University Press called New Mathematical Monographs.
		The first sentence of Chapter I, for example, is "The main functors in representation theory of finite groups are the restriction to subgroups and its adjoint, called induction." That sends, I think, the correct signal about the authors' approach: we're doing serious work with serious pre-requisites.
		Representation Theory of Finite Reductive Groups, by Marc Cabanes and Michel Enguehard.
	www.maa.org /reviews/brief_jan05.html (1873 words)

	Amazon.co.uk: Books: Representation Theory: A First Course (Graduate Texts in Mathematics S.) (Site not responding. Last check: 2007-11-06)
		The primary goal of these lectures is to introduce a beginner to the finite-dimensional representations of Lie groups and Lie algebras.
		The general theory is developed sparingly, and then mainly as useful and unifying language to describe phenomena already encountered in concrete cases.
		The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups.
	www.amazon.co.uk /exec/obidos/ASIN/0387974954 (418 words)

	CREP Home Page
		CREP is the abbreviation for Combinatorial REPresentation theory.
		Additional material on computational aspects in the representation theory of algebras can be found in the references of the paper by P. Dräxler and R. Nörenberg.
		BIREP (:= Representation theory of algebras at Bielefeld)
	www.mathematik.uni-bielefeld.de /~sek/crep.html (743 words)

	A Representation Theory for Morphological Image and Signal Processing (Site not responding. Last check: 2007-11-06)
		A unifying theory for many concepts and operations encountered in or related to morphological image and signal analysis is presented.
		This approach leads to a general representation theory, in which any translation-invariant, increasing, upper semicontinuous system can be presented exactly as a minimal nonlinear superposition of morphological erosions or dilations.
		The theory is used to analyze some special cases of image/signal analysis systems, such as morphological filters, median and order-statistic filters, linear filters, and shape recognition transforms.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tp/&toc=comp/trans/tp/1989/06/i6toc.xml&DOI=10.1109/34.24793 (958 words)

	Algebraic Representation Theory
		We are interested in the representation theory of algebras and groups arising in Lie Theory.
		The representation theory of such objects over a field of positive characteristic (or in the quantum case, when certain parameters are roots of unity) is not well-understood.
		Determining the structure of such modules is in general difficult; however, in certain cases there are algorithms to do so.
	www.city.ac.uk /sems/mathematics/research/algebra/algebraicrep.html (172 words)

	Ventral/dorsal, predicate/argument: the transformation from perception to meaning. (Site not responding. Last check: 2007-11-06)
		It is necessary to distinguish among representations caused directly by perception, representations of past perceptions in long-term memory, the representations underlying linguistic utterances, and the surface phonological and grammatical structures of sentences.
		A theory of the input-output mappings of a particular programmed computer might not need to get down to the 1s and 0s, but if the mappings are at all complicated, some quite abstract underlying representations, no doubt resembling the computers' program(s), would be required.
		Perception of motion and mental representation of motion properties are at present probably the most problematic area for the central claim of the target article, and clearly more research, and perhaps some revision of the central claim, is necessary.
	www.ling.ed.ac.uk /~jim/bbsreply5.html (9761 words)

	Loeks Representation Theory Page
		In the case of real symmetric spaces this has been a major area of research in the last few decades and was only recently completed by Delorme.
		Another area of representation theory where symmetric varieties play a role is that of the so-called canonical representations.
		The next challenge is to see which unitary representations can be produced with this procedure.
	www4.ncsu.edu /~loek/research/repr.html (417 words)

	BBSPrints Archive: The emulation theory of representation: motor control, imagery, and perception
		The emulation theory of representation is developed as a framework for understanding representational capacities of the brain.
		On this theory, the brain constructs emulators the body and environment which, during normal sensorimotor behavior, are run in parallel with the emulated systems in order to produce expectations and otherwise enhance sensory information.
		The emulation theory of representation, is developed and explored as a framework that can revealingly synthesize a wide variety of representational functions of the brain.
	bbsonline.cup.cam.ac.uk /documents/a/00/00/11/90 (419 words)

	[No title]
		In the 1980's, results from the theory of automorphic forms were used to construct explicit families of Ramanujan graphs, that is, graphs for which Laplace eigenvalues satisfy strong inequalities.
		The theory of arithmetic groups and representation theory of semisimple groups have led to results on the diameters and expansion properties of finite simple groups.
		While there is now a proof of this independent of representation theory, the only known proof for specific "natural" sets of generators, even for SL(2,p), relies on the celebrated Selberg Theorem on the eigenvalues of the Laplacian on arithmetic hyperbolic groups.
	www.ipam.ucla.edu /programs/agg2004 (814 words)

	[No title]
		I participated also at the seminar of Number Theory lead by Professor N. Popescu devoted to algebraic number theory, analytic number theory, class field theory and theory of algebraic functions.
		Fields of interest: Number theory with special interest in analytic number theory, algebraic numbers and functions, transcendental numbers, local class field theory, diofantine equations.
		On the existence of trace for elements of $\CC_p$, Algebras and Representation Theory, to appear.
	www.imar.ro /~mvajaitu (942 words)

	Representation Theory of Lie Groups - IMS (Site not responding. Last check: 2007-11-06)
		Representation Theory of Lie Groups is the systematic study of symmetries and ways of exploiting them.
		For the Local Langlands Conjecture, the current proof involves tools from number theory and algebraic geometry and represent a new set of promising ideas that may continue to bear fruits in the near future.
		On the other hand, many interesting representations of Lie groups are often multiplicity-free as representations of other groups such as the maximal compact subgroups.
	www.ims.nus.edu.sg /Programs/liegroups (481 words)

	Robert Langlands' work - main page (Site not responding. Last check: 2007-11-06)
		He has held faculty positions at Princeton University and Yale University, and is currently a Professor at the Institute for Advanced Study in Princeton.
		He has won several awards recognizing his outstanding contributions to the theory of automorphic forms, among them an honorary degree from the University of British Columbia in 1985.
		Representation theory---its rise and its role in number theory
	www.sunsite.ubc.ca /DigitalMathArchive/Langlands/intro.html (530 words)

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