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Topic: Reducible representation

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	PlanetMath: a representation which is not completely reducible
		"a representation which is not completely reducible" is owned by bwebste.
		Cross-references: complementary subspace, spans, spanned by, generates, identity, characteristic function, homomorphism, subspace, invariant subspace, complementary, constant functions, subrepresentation, obvious, completely reducible, representation, action, functions, regular representation, Maschke's theorem, group, order, divide, characteristic, field, finite group
		This is version 2 of a representation which is not completely reducible, born on 2003-03-24, modified 2005-03-10.
	planetmath.org /encyclopedia/ARepresentationWhichIsNotCompletelyReducible.html (170 words)

	The first few HW's will be kind of easy due to the short weeks arising from the introductory lecture & labor day, ... (Site not responding. Last check: 2007-10-11)
		Decompose this reducible representation into its irreducible components and then from inspecting the character tables compose all of the possible combinations of s, p, d, f orbitals that can match that bonding arrangement.
		Once you get your reducible representation decomposed use the type and number of each irreducible representation as a heading for a table of possible exact representations.
		The reducible representation, the table of exact representations and the final hybrid orbitals are all required components for this question.
	www.uvm.edu /~swgordon/hw5.html (1310 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.
		Any representation of a group also gives a representation of a subgroup, but an irreducible representation of the group may give a reducible representation of a subgroup.
		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)

	Irreducible Representation -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.
		In a given representation, reducible or irreducible, the group characters of all matrices belonging to operations in the same class are identical (but differ from those in other representations).
		The number of irreducible representations of a group is equal to the number of conjugacy classes in the group.
	mathworld.wolfram.com.cob-web.org:8888 /IrreducibleRepresentation.html (311 words)

	Symmetry and Bonding (Site not responding. Last check: 2007-10-11)
		The number should come out as zero or as a positive integer: if it does not then either the formula has been used incorrectly or the reducible representation has been generated incorrectly...or both.
		The representations for the axial and equatorial hybrids are:
		Here the reduction will be performed on the representation of the 5 hybrids although it could be done on the two sets separately.
	www.hull.ac.uk /php/chsajb/symmetry/reduction.html (798 words)

	[No title]
		Given a field $K$ and a representation $\rho$ of a group to $SL(2,K)$, the \it character \rm of the representation sends a group element $g$ to the trace of the matrix $\rho (g)$.
		Given a reducible representation $\rho$ of a group to $SL(2, K)$, we may assume after conjugate $\rho$ by an $SL(2,K)$ matrix that the image of $\rho$ is in the set of upper triangular matrices.
		In particular, the representation $\rho$ induces a representation $\rho'$ of $G$ to $SL(2,K)$ whose character is $f$.
	www.intlpress.com /journals/JDG/archive/vol.54/issue3/luo-dir/luo (9127 words)

	[No title]
		Below the character table, a section of the template will be devoted to the generation of the reducible representation for a particular molecule under investigation.
		Below this the irreducible representation is created and the rotational and translational spectrum terms are removed to produce the vibrational terms.
		The reducible representation is required in making the calculations for the multiplier for irreducible representation that will appear in the blank lines of the character table.
	www.cbu.edu /~mcondren/SPRDSHT.htm (1475 words)

	Perception, Representation, and Reference
		In some cases the knowledge to be reduced may not have been explicated to the level of detail of the reducing knowledge.
		It may be argued that this form of reference depends upon the internal representation of the objects to be selected, and the problem of representation is the other side of the same coin of which reference is cast.
		Symbols are "about" their referents; representations are "about" what they stand for; and perceptions are "about" that which gives rise to them (whether in physical or semantic environments).
	www.xenodochy.org /article/mathesis.html (14238 words)

	Abelian/non-Abelian, and the U(1), SU(2), SU(3) Groups
		Matrix Representation - This representation is in the form of matrices.
		The dimension of a representation is the dimension of the space on which it acts - the above example is 2 dimensional.
		The condition for a solution is the vanishing of the determinant of the coefficient of the v's, and this reduces to: p =
	universe-review.ca /R15-10-groups.htm (3565 words)

	5 rep and 10 rep etc...
		If the representation is reducible, the OBJECT is composite, in the sense that it can be broken into more elementary objects belonging to irreducible representation.
		I'm not sure what the mathematical requirements are for which reps are reducible and which are not, but for the usual goups we deal with, I think that reducible groups can always be put into irreducible form.
		Sir, the matter we see in nature is reducible to its irreducible and fundamental parts, fermions and bosons.
	www.physicsforums.com /showthread.php?t=91770 (2188 words)

	Groups for Dummies
		Now the reducible representation produced by the basis x,y,z is B + E. Consider 3 objects in a row, labelled 1,2 and 3.
		What we are looking for is choices of bases for representations that reduce them to the simplest level, and these are the bases of the irreducible representations.
		It is no accident that it also has three irreducible representations, since it happens that the number of inequivalent irreducible representations is equal to the number of classes, as we shall prove.
	www.du.edu /~jcalvert/math/groupdum.htm (7838 words)

	SCH330 - 8. Symmetry of molecules (Site not responding. Last check: 2007-10-11)
		Now find the trace of the reducible representation from the basis of the three co-ordinate vectors on each of the 7 atoms.
		Reduce this representation and hence find the symmetry of the normal vibration modes.
		Thus these two translations remove one of the E representations found when you reduce the reducible representation.
	www.chem.swin.edu.au /courses/ntu/sch330/exp8.html (343 words)

	Springer Online Reference Works
		Reducible representation); or b) for each rational linear representation
		However, even in the general case, a reductive group can be described in terms of its representation theory.
		A linear algebraic group is reductive if and only if it is geometrically reductive (see Mumford hypothesis).
	eom.springer.de /r/r080440.htm (379 words)

	Ch231 - HW 6 (Site not responding. Last check: 2007-10-11)
		Find a reasonable point group for :SF and copy just the name of the group, the operators (including class multipliers), the names of the group representations and their characters onto a sheet of paper with plenty of room underneath the last line.
		Draw a horizontal line under the last representation of the character table and place the characters for the reducible representation for just the bonds,
		d) Find the number of IR active CO stretches by generating a reducible representation for the CO stretches using double ended arrows localized on the CO bonds and then reducing it into irreducible components.
	www.uvm.edu /~swgordon/hw7.html (974 words)

	Realms of Cognitive Science (Site not responding. Last check: 2007-10-11)
		It appears that there are three realms: the realm of matter, the realm of representations, and the realm of qualia/intentions/consciousness, not just two: matter and consciousness.
		Modern cognitive science holds that the representational realm is reducible to the functional realm, and I agree, although some, e.g.
		Penrose may in some strange sense hold that consciousness is reducible to function, but believes that the right function can only be realized through quantum effects (but he's so vague that one can't know).
	consc.net /notes/realms.html (651 words)

	Research Training Group in Representation Theory, Geometry and Combinatorics - UC Berkeley Mathematics (Site not responding. Last check: 2007-10-11)
		Three-year postdoctoral positions funded jointly by NSF and the university, with a reduced teaching load of one course per semester, support for summer research in each of the first two years, and additional funds for travel and other research-related expenses.
		Some of them are: representation theory of infinite dimensional Lie algebras and quantum groups, combinatorics of weight multiplicities, invariants of knots and 3-manifolds, geometry of moduli spaces of flat G-bundles over surfaces, etc. This is roughly the direction of my research.
		In recent years his research has been concentrated on the interplay between the theory of representations of semisimple Lie groups and the geometry of complex flag manifolds.
	math.berkeley.edu /research_rtg.html (761 words)

	GAP (repsn) - Chapter 3: Reducible Representations
		In this chapter we introduce some functions which deal with a complex reducible representation R of a finite group G. 3.1 Constituents of Representations
		is a reducible representation of G. 3.2 Block Representations
		is a reducible representation of a group G, this function returns a block diagonal representation of G equivalent to
	www-groups.dcs.st-and.ac.uk /~gap/Manuals/pkg/repsn/htm/chap3.html (175 words)

	Universal Press Syndicate: Submissions
		Usually, comic strips and panels are drawn twice the size that appears in your local newspaper.
		Any size is acceptable as long as it can be reduced proportionately.
		Strips should be reducible to 13 by 38.4 picas; panels to 24 by 19 picas.
	www.amuniversal.com /ups/submissions.htm (510 words)

	[No title]
		Decompose a 3P electronic state into its irreducible representations in a crystal field of D4h symmetry, (a) ignoring coupling of the electron spins to the crystal field (b) including the coupling of the electron spin to the orbital angular momentum and hence to the crystal field 24.
		Use the representation matrices of C3v determined in problem 10 to determine all the antisymmetric wavefunctions for two electrons in an e type orbital.
		Determine the irreducible representations of the double group of C3v spanned by this four dimensional representation.
	www.jhu.edu /~chem/030345/Homework.doc (1388 words)

	Ammonia
		T and R show the representations to which components of the translation and rotation displacements belong; these are vectors and axial vectors, respectively.
		T also shows the representations of the dipole moment operator which produces the infrared spectrum.
		Therefore, the characters of the reducible representation of the displacements is 12, 0, 2.
	www.du.edu /~jcalvert/phys/ammonia.htm (1876 words)

	JCE 1997 (74) 809 [Jul] Reducible Representation for Linear Molecules
		JCE 1997 (74) 809 [Jul] Reducible Representation for Linear Molecules
		Simple methods are described that permit the ready writing of reducible representations for systems described by the point groups D
		Examples of applying the methods are given to problems in vibrational and electronic spectroscopy as well as bonding theory.
	jchemed.chem.wisc.edu:8000 /Journal/Issues/1997/Jul/abs809.html (165 words)

	UC Davis Math: Prelim Exam Preparation
		Find all endomorphisms (or all automorphisms) of a representation decomposed into a sum of irreps.
		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: 2007-10-11)
		MAGE display used to show determination of reducible representation for total modes of motion, in water molecule.
		Total modes of motion include: translational + rotational + vibrational modes.
		So, the reducible representation for total modes of motion in water molecule has characters as shown in the table:
	www1.luc.edu.cob-web.org:8888 /faculty/spavko1/JCE/Mage/mom.htm (286 words)

	JCE 2003 (80) 1380 [Dec] Vibrational Analysis for C₆₀ and Other Fullerenes
		I would like to supplement their presentation with another approach, in widespread use, which records the number of atoms that are unmoved by each symmetry operation, yielding the reducible representation Γ
		, the reducible representation for the 180 degrees of freedom of the C
		irreducible representations by the usual method as shown in Table 1 of the Supplementary Materials.
	jchemed.chem.wisc.edu /Journal/issues/2003/Dec/abs1380_1.html (516 words)

	[No title]
		If it is not an integer then something went wrong either with your arithmetic or, more likely, with the determination of the characters of your reducible representation!!
		We will assume that such a set of translations can be formed and we will make it the basis for a representation in the point group of a tetrahedron.
		So we take the directional properties of the four translation vectors t1, t2, t3 and t4 as a basis for the representation of the point group Td.
	www.chem.gla.ac.uk /~lutz/molecsym/mosylec5m.doc (615 words)

	Interaction of Radiation and Matter - Spectroscopy (Site not responding. Last check: 2007-10-11)
		Number of Irreducible Representations in the Reducible Representation
		Example of Coordinate Transformation: Reducible Representation Matrix Cs Formaldehyde
		Number of Irreducible Representations in the Reducible Representation Cs Formaldehyde
	www.chem.brown.edu /chem277/lect12.html (328 words)

	Elementary Group Theory (Site not responding. Last check: 2007-10-11)
		Example of a triangular planar molecule (AB) point group D
		Matrix Representation for symmetry elements and corresponding operators.
		Decomposition of a reducible representation into irreducible representations 23
	www.vpagrawal.com /Elementary%20Group%20Theory.htm (1423 words)

	math_tools class Reference (Site not responding. Last check: 2007-10-11)
		Projects out irrep components from a reducible representation.
		the matrix of representations, each representation should be a row of this matrix whose length = number of irreps Parses input
		Performs bfgs update on inverse of hessian matrix.
	zopyros.ccqc.uga.edu /~kenny/extrema-doc/classmath__tools.html (334 words)

	CSERD: Catalog (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-10-11)
		Description: This site has most chemically relevant character tables in electronic form.
		A useful utility automatically finds the irreducible representation from any reducible representation.
		This item has 1 review available (1 verification; 0 validation; 0 accreditation):
	cserd.nsdl.org.cob-web.org:8888 /Catalog/Item/625 (288 words)

	Class Summary
		5) Solve G to find irreducible representations by Projection Operators
		: character of the n operation in G, the reducible representation
		The number of orbitals with symmetry type m in G is given by
	www.chem.purdue.edu /wenthold/chm668/Dailies/102704.htm (144 words)

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