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	Group representation - Wikipedia, the free encyclopedia
		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.
		A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V.
	en.wikipedia.org /wiki/Group_representation (1500 words)

	Adjoint representation - Wikipedia, the free encyclopedia
		In mathematics, the adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation.
		The representation is equivalent to that given by the action of G by linear substitution on the space of binary (i.e., 2 variable) quadratic forms.
		According to the philosophy in representation theory known as the orbit method, the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits.
	en.wikipedia.org /wiki/Adjoint_representation (415 words)

	Fundamental representation - Wikipedia, the free encyclopedia
		All other irreducible representations of the group can be found in the tensor products of the fundamental representation with many copies of itself.
		For example, the fundamental representation of SO(n) or SU(n) are n-dimensional vector spaces, and the fundamental representation of E8 is 248-dimensional.
		Another approach would be to specify that the matrix coefficients of the representation are a sufficiently rich set of functions on the group.
	en.wikipedia.org /wiki/Fundamental_representation (240 words)

	Group White Sheet FAQ
		Faithful representations are permutation representations (homomorphisms from a group
		The important consequence of faithful representations is that the original group is embedded as a subgroup of the symmetric group.
		Transitive representations are permutation representations whose image is a transitive group.
	darkwing.uoregon.edu /~jwilson7/math/WhiteSheetFAQ.html (795 words)

	Agreements with Hell and Other Objects of Our Faith-- Part II
		Thus, to have fidelity is to be faithful - literally, to be full of faith, full of confidence in the value of that which we are faithful to.
		To be faithful is to be faithful even though one does not know whether the other will live up to the other's obligations.
		For a politician to say that she is not faithful to the Constitution is treasonous; rather she must say that she has faith in the deeper political principles that underlie the document and remain even to this day its nourishing source and fountainhead.
	www.yale.edu /lawweb/jbalkin/articles/agrhell2.htm (8499 words)

	[No title]
		A three-dimensional faithful representation dim3b that consists of all the symmetries of the regular tetrahedron.
		A three-dimensional faithful representation dim3athat consists of all the rotations of a cube.
		A four-dimensional faithful representation dim4 that consists of all the "rotations" of the regular simplex in four-space.
	mathlab.cit.cornell.edu /visualization/tenseg/tenseg.html (1441 words)

	GT Vol 3 (1999) Paper 16 (Abstract)
		The Burau representation is not faithful for n = 5
		The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1.
		It is a longstanding open problem to determine for which values of n this representation is faithful.
	www.maths.warwick.ac.uk /gt/GTVol3/paper16.abs.html (116 words)

	Group representation (Site not responding. Last check: 2007-10-21)
		Representation theory is important because it enables many group-theoretic problems to be reduced to problems in linear algebra, which has a very well-understood theory.
		A faithful representation is one in which the homomorphism G → GL(V) is injective.
		Representations of finite groups can always be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem.)
	www.free-download-soft.com /info/dialers.html (1407 words)

	PlanetMath: Lie algebra representation
		A representation is called irreducible or simple if its only invariant subspaces are
		Given a pair of representations, we can define a new representation, called the direct sum of the two given representations:
		This is version 13 of Lie algebra representation, born on 2002-05-29, modified 2005-08-05.
	planetmath.org /encyclopedia/Representation2.html (131 words)

	Garr & Schlueter, Ltd. \| Our Philosophy (Site not responding. Last check: 2007-10-21)
		Legal counsel and representation during these complex times is crucial, to safeguard not only our own personal and business decisions, but also to safeguard us from the personal and business decisions of others.
		The approach is simple: faithfully and honestly build a client relationship so that strenuous representation of the client's interests is achieved, always keeping in mind how we ourselves at Garr & Schlueter, Ltd. would prefer to be treated and represented if we were in our client's shoes.
		Honesty, faithful representation, service and an understanding of trust are the basis of the Garr & Schlueter, Ltd. approach.
	www.gslawoffices.com /philo.html (302 words)

	Stephen Bigelow (Site not responding. Last check: 2007-10-21)
		A 64-dimensional faithful representation of the mapping class group of a genus 2 surface.
		The Lawrence-Krammer representation is a faithful representation of the braid group into a general linear group over the reals.
		The Burau representation is not faithful for n=5
	www.math.ucsb.edu /~bigelow/papers.html (289 words)

	Paper on representation of the Reals (and others from Darmstadt) (Site not responding. Last check: 2007-10-21)
		``Faithful'' refers to the fact that we overcome the classical problem of multiple representations for certain numbers.
		Hence we do also give a faithful representation of the topological structure of the real numbers.
		The quasi-uniformity on our model may be described in a finitary fashion: There is a base consisting of relations U_n such that in order to determine whether x U_n y holds, one needs to know only the first n digits of the sequences x and y.
	www.cis.upenn.edu /~bcpierce/types/archives/1994/msg00010.html (291 words)

	[tex-fonts] Re: Strange cmsy10.pfb... (Site not responding. Last check: 2007-10-21)
		Faithful means faithful to the human eye; so the meaning of faithful has both physical and intellectual facets of varying importance.
		(c) Where arcs have to be split in order to obtain a faithful representation, insist that the splitting points be on the parent arc(s) and that the directions there be be preserved.
		It is the 'faithful representation' of (parts of) elliptic pen envelopes by bezier cubics that forces the above subtle approximation problems upon us.
	www.math.utah.edu /pipermail/tex-fonts/2003-June/000443.html (1170 words)

	AGT 1 (2001) Paper 34 (Abstract)
		In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers.
		We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the close relationship between this group and B_{n-1}.
		We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hilden's result that this group is a Z_2 central extension of the mapping class group of the 6-punctured sphere.
	www.maths.warwick.ac.uk /agt/AGTVol1/agt-1-34.abs.html (183 words)

	Research Articles - Julianne Geering Rainbolt
		Abstract: In this paper we explicitly calculate the irreducible representations of the endomorphism algebra of the Gelfand-Graev representation of the unitary group U(3,q).
		One is the regular representation and one is the usual Gelfand-Graev representation.
		The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.
	euler.slu.edu /Dept/Faculty/rainbolt/articles.htm (482 words)

	Austin-Sparks.Net - Titles of the Risen Lord
		Three titles: the faithful Witness, the Firstborn of the dead, the Ruler of the kings of the earth.
		If we understood all that was going on in this necessity for faith: one Who had, through past eternity, never had to live by faith at all – never had to live by faith.
		He made no pretence that man is better than he is. He was faithful in His Witness as to what man is. We might be afraid to do that.
	www.austin-sparks.net /english/000418.html (4862 words)

	BACKGROUND
		There are two canonical representations of braid groups by matrices over Laurent polynomial rings - the Burau and Gassner representations (the latter is actually a representation of the pure braid group which is a subgroup of finite index in the whole braid group).
		The Burau representation was shown to be non-faithful for n \ge 10 in [J.Moody, The faithfulness question for the Burau representation, Proc.
		The "archimedian" superrigidity means that for each representation h: \Gamma \to GL_m(R) which has the image that is not relatively compact, there exists a finite index subgroup \Gamma' \subset \Gamma such that the restriction h\Gamma' is induced by a representation of the entire automorphism group of HH^n.
	www.cs.gc.cuny.edu /~cryptlab/gworld/problems/Back2.html (4773 words)

	FASB: Conceptual Framework
		Faithful representation of real-world economic phenomena is an essential qualitative characteristic, which includes capturing the substance of those economic phenomena.
		Representations are faithful—there is correspondence or agreement between the accounting measures or descriptions in financial reports and the economic phenomena they purport to represent—when the measures and descriptions are verifiable, and the measuring or describing is done in a neutral manner.
		Materiality should be included in the converged framework as a screen or filter to determine whether information is sufficiently significant to influence the decisions of users in the context of the entity, rather than as a qualitative characteristic of decision-useful financial information.
	www.fasb.org /project/conceptual_framework.shtml#history (2229 words)

	\Large {The Mathematics Survey Proposal}
		Summary This is a proposal to construct a new means of organizing, communicating and archiving mathematical knowledge, by a faithful representation of that knowledge in cyberspace.
		The proposal is to construct a new means of organizing, communicating and archiving mathematical knowledge, by a faithful representation of that knowledge in cyberspace.
		For that webpage will be the most visible and conspicious representation of that branch of knowledge in cyberspace: a professionally organized, peer-reviewed guide to what is currently regarded as important in that branch of knowledge, with links to other such guides in related branches.
	www.stat.berkeley.edu /users/pitman/mathsurvey/proposal.html (4429 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)

	[No title]
		Also, for this same surface (, we exhibit another discrete faithful representation (: (1(() (PU(2,1) such that the limit set of the group (=(((1(()) (PU(2,1) is a tame knot.
		It can also be shown that in this space, geometrically finite quasifuchsian representations are structurally stable, therefore, any small type-preserving deformation of such representation has the same topological properties.
		In particular, any small deformation of such representation (: G (PU(2,1) preserves the topological type of the quocient M(((G)) = R(((G))/((G), where R(((G)) is the regular set of ((G).
	www.ufmg.br /prpg/dow_anais/cien_ex_terra/matematica_3/fdutenhefner_abs.doc (227 words)

	MSN Encarta - Realism (art and literature)
		The novelist George Eliot introduced realism into English fiction; as she declared in Adam Bede (1859), her purpose was to give a “faithful representation of commonplace things.”Mark Twain and William Dean Howells were the pioneers of realism in the United States.
		In general, the work of these writers illustrates the main tenet of realism, that writers must not select facts in accord with preconceived aesthetic or ethical ideals but must set down their observations impartially and objectively.
		Concerned with the faithful representation of life, which frequently lacks form, the realists tended to downplay plot in favor of character and to concentrate on middle-class life and preoccupations, avoiding larger, more dramatic issues.
	encarta.msn.com /encyclopedia_761552472/Realism_(art_and_literature).html (552 words)

	Braid Groups are Linear - Bigelow (ResearchIndex)
		A group is said to be linear if it admits a faithful representation into a group of matrices over R. Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case n = 4.
		0.4: The Burau Representation is Not Faithful for n = 5 - Bigelow (1999)
		3 The burau representation is not faithful for n - Bigelow - 1999
	citeseer.ist.psu.edu /bigelow01braid.html (550 words)

	GIPS Preprint 2005-001 (Site not responding. Last check: 2007-10-21)
		We prove that it is isospectral and that it is endowed with a Poisson structure, and discuss its invariants and Casimirs.
		Formulation of the Poisson problem in a Lie—Poisson setting, as a flow on a dual of a Lie algebra, requires a computation of its faithful representation.
		Although the existence of a faithful representation is assured by the Ado theorem and a symbolic algorithm, due to de Graaf, exists for general computation of faithful representations of Lie algebras, the practical problem of forming a ‘tight’ representation, convenient for subsequent analytic and numerical work, belongs to numerical algebra.
	www.focm.net /gi/gips/2005/001.html (140 words)

	SET THEORY, QUANTUM SET THEORY & CLIFFORD ALGEBRAS
		A faithful but reducible representation exists; it exists as a direct sum of two irreducible (nonfaithful) representations where the direct summands are not only subalgebras, but also mutually annihilating two-sided ideals in the faithfully representing algebra of dimension 2
		The faithful representation is, of course, not simple, by virtue of it being a direct sum.
		For n = 3, the faithful representation is expressed by the direct (reducible) sum M(2) + M(2) = CL(3), which is a subalgebra of M(4).
	graham.main.nc.us /~bhammel/QSET/qset1.html (10320 words)

	[No title]
		To qualif y as a transaction, the effect of the underlying events on the business\rquote financial statements must be subject to reliable measurement.
		A reliable measurement is a description in words and numbers that is reasonably free from error and bias and tha t is a faithful representation of what it claims to represent.
		The transfer of the building and equipment to the business from Susan Eel, the owner of the business, howe v er, is not an exchange between the business and an outside party; thus, its amount may be biased and a less than faithful representation of fair value of the building and equipment.
	www.cob.ohio-state.edu /~ramamurthy_3/rr211/homework/hw3.doc (3215 words)

	Ephilosopher :: Political Philosophy Forum :: Representation (Site not responding. Last check: 2007-10-21)
		for an MP or congressman)-- is it faithful representation of constituents' positions and opinions, or is it acting in the best interests of constituents given the information one has (which can at times mean acting against constituents' views).
		Say, for example, that you could only have one (faithful representation of one's views or, exclusively, a representative who acts in what s/he believes is in your best interest given the information that s/he has).
		As far as accurate representation no matter how ignorant versus using professional knowledge to pursue best interests, I have serious problems with a system that chooses either one.
	www.ephilosopher.com /phpBB_14-action-viewtopic-topic-68.html (2309 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In many cases the approximation will be a faithful representation of the path set defined by the strategy (i.e., the old propagation graph which AP Studio already does).
		If the representation is not faithful, a message should be given.
		You need a consistency checker for this which is already implemented in Demeter/C++ for series-parallel graphs.
	www.ccs.neu.edu /research/demeter/design-decisions/traversal-strategies/traversal_graph_notation (426 words)

	Institute of Chartered Accountant of India (Site not responding. Last check: 2007-10-21)
		Thus, for example, a balance sheet should represent faithfully the transactions and other events that result in assets, liabilities and equity of the enterprise at the reporting date which meet the recognition criteria.
		Most financial information is subject to some risk of being less than a faithful representation of that which it purports to portray.
		This is not due to bias, but rather to inherent difficulties either in identifying the transactions and other events to be measured or in devising and applying measurement and presentation techniques that can convey messages that correspond with those transactions and events.
	www.icai.org /resource/fr15.html (200 words)

	Faithfulness -- KEVIN CRAIG for Congress
		Be thou faithful unto death, and I will give thee a crown of life.
		Now faith is the substance of things hoped for, the evidence of things not seen.
		My life is a faithful representation of God's blueprints for man. My allegiance to God's Law and my strict adherence to duty leads others to admire God and His Commands.
	members.aol.com /craig4congress/character/faithful.htm (271 words)

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