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Topic: Irreducible representation

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	Irreducible Representation -- from Wolfram MathWorld
		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 /IrreducibleRepresentation.html (303 words)

	Irreducible (mathematics) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In (Click link for more info and facts about representation theory (group theory)) representation theory (group theory), an irreducible representation is a nontrivial (A creation that is a visual or tangible rendering of someone or something) representation with no nontrivial subrepresentations.
		The notions of irreducibility in algebra and manifold theory are related.
		An irreducible manifold is thus prime, although the converse does not hold.
	www.absoluteastronomy.com /encyclopedia/i/ir/irreducible_(mathematics).htm (537 words)

	Java: Lie Group Representations
		Representations of the universal cover of SL(2,R) can be realized as global sections of standard weakened Harish-Chandra sheaves on the flag variety of the group PSL(2,C).
		A module is irreducible when it is nonzero and has no proper submodules; otherwise the module is trivial (0) or reducible.
		Nevertheless, there are irreducible representations of G supported on closed orbits.
	www.panix.com /~shalla/java/sl2r.html (762 words)

	Group representation: Definition and Links by Encyclopedian.com - All about Group representation (Site not responding. Last check: 2007-10-29)
		In abstract algebra, a representation of a finite group G is a group homomorphism from G to the general linear group GL(n,C) of invertible complex n-by-n matrices.
		This representation is said to be faithful, because ρ is a one-to-one map.
		The characters of all the irreducible representations of a finite group form a character table, with conjugacy classes of elements as the columns, and characters as the rows.
	www.encyclopedian.com /gr/Group-representation.html (641 words)

	Encyclopedia: Modular representation
		In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K divides the order of G.
		Modular representations are very different from when K is the complex numbers, or when the characteristic of K does not divide the order of G.
		By contrast, in the non-modular case every irreducible representation is a direct summand in the regular representation, implying that it is projective.
	www.nationmaster.com /encyclopedia/Modular-representation (414 words)

	Science Fair Projects - Fundamental representation
		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.
		In greater generality it is not expected to happen in the way that all irreducible representations occur as direct summands; rather (at best) as subquotients.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Fundamental_representation (390 words)

	Irreducible Info - Bored Net - Boredom (Site not responding. Last check: 2007-10-29)
		In abstract algebra, irreducible is an abbreviation of irreducible element; for example an irreducible polynomial.
		In representation theory, an irreducible representation is one with no nontrivial subrepresentations.
		In algebraic geometry, an irreducible algebraic variety W is one that cannot be written as a union of subvarieties U and V, except when one of those is contained in the other.
	www.borednet.com /e/n/encyclopedia/i/ir/irreducible.html (264 words)

	S1SN, S2SN, S3SN - Online Information article about S1SN, S2SN, S3SN (Site not responding. Last check: 2007-10-29)
		When this cannot be done the group is called " irreducible." It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves.
		This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.
		The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following.
	encyclopedia.jrank.org /RON_SAC/S1SN_S2SN_S3SN.html (2493 words)

	Irreducible (mathematics) - Wikipedia, the free encyclopedia
		In representation theory, an irreducible representation is a nontrivial representation with no nontrivial subrepresentations.
		In the theory of manifolds, an n-manifold is irreducible if any embedded (n−1)-sphere bounds an embedded n-ball.
		A matrix is irreducible if it cannot be made upper triangular via a matrix permutation.
	en.wikipedia.org /wiki/Irreducible_(mathematics) (359 words)

	Casimir invariant - Wikipedia, the free encyclopedia
		The number of independent elements of the center of the universal enveloping algebra is also the rank in the case of a semisimple Lie algebra.
		In any irreducible representation of the Lie algebra, by Schur's Lemma, any member of the center of the universal enveloping algebra commutes with everything and thus is proportional to the identity.
		This constant of proportionality can be used to classify the representations of the Lie algebra (and hence, also of its Lie group).
	www.wikipedia.org /wiki/Casimir_operator (166 words)

	[No title]
		The 10-dimensional representation is the adjoint representation and its weights are the roots $\{2\a_1+\a_2\,, \,\a_1+\a_2\,, \,\a_1\,, \,\a_2\,, \,0\,, \, 0\,,\,-\a_2\,, \,-\a_1\,, \newline -\a_1-\a_2\,, \,-2\a_1-\a_2\}$.
		The 14-dimensional representation is the adjoint representation with weights equal to the roots $\{2\a_1+3\a_2\,, \,\a_1+3\a_2\,, \,\a_1+2\a_2\,, \,\a_1+\a_2\,, \,\a_1\,, \a_2\,,\, 0\,, 0\,, \,-\a_2\,, \,-\a_1\,, \,-\a_1-\a_2\,, \,-\a_1-2\a_2\,, \, -\a_1-3\a_2\,, \,-2\a_1-3\a_2\}$.
		The irreducibility of $\pi^\prime$ follows immediately from the two-way connectedness of $G^\prime$.} Applying this theorem to the example in figure \ref{redgraph} we see that there is a 15-dimensional irreducible representation of \gqh.
	www.ma.utexas.edu /mp_arc/papers/94-128 (5529 words)

	Representation theory of SU(2) - Enpsychlopedia (Site not responding. Last check: 2007-10-29)
		In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups.
		By Schur's lemma, it is proportional to the identity for irreducible representations.
		A heighest weight representation is a representation with a weight α which is greater than all the other weights.
	www.grohol.com /psypsych/Representation_theory_of_SU%282%29 (480 words)

	Outline
		When determining the dimension of all irreducible representations of the group of the Schrödinger equation from the character table all degrees of degeneracy in the system that belongs to this group are found.
		Example: an irreducible representation A1g is one-dimensional, and the functions for which this irreducible representation forms a basis are symmetric with respect to inversion, symmetric with respect to rotation about Cn, and symmetric with respect to reflection in sv..
		In a cartesian representation of the point group a cartesian coordinate system is applied to each atom, and the transformation of the coordinates by the symmetry operations of the point group is considered.
	w3.rz-berlin.mpg.de /~horn/SemiconductorSurfaces/Symmetry-HTML/Symmetrie98.html (2027 words)

	Representation Theory (Site not responding. Last check: 2007-10-29)
		Compute the absolutely irreducible representations of the group G over appropriate extensions or sub-fields of the given field k.
		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.
	www.mat.niu.edu /help/math/magmahelp/text346.html (789 words)

	Springer Online Reference Works
		The representation contragredient to a rational representation is a rational representation.
		is finite, then each of its linear representations will be a rational representation, and the theory of rational representations coincides with the theory of representations of finite groups (cf.
		To a large extent, specific methods of the theory of linear algebraic groups are used to study rational representations in case the group under consideration is connected, and the most thoroughly developed theory is that of rational representations of connected semi-simple algebraic groups.
	eom.springer.de /R/r077630.htm (501 words)

	[No title]
		We consider unitary representations of Γ which are weakly contained in the regular representations.
		There are lots of representations of Γ, and one's intuition is that a "generic" representation is irreducibile.
		In certain cases one can prove simultaneously that a representation is irreducible and that it has exactly two inequivalent, irreducible boundary realizations.
	www.ipam.ucla.edu /abstract.aspx?tid=3872 (294 words)

	The representation theory of #tex2html_wrap_inline1613#
		for the three dimensional irreducible representation comes from the fact that, in general algebra, such modules are called ``Steinberg modules''.
		One can, for instance, study the projective covers of the different representations (for completeness sake, this information is represented by dashed lines on figure 1), the subfactor representations, the quiver of the algebra, its Cartan matrix etcThis, however, would be a bit technical and more appropriate for a review paper (see [10]).
		Irreducible representations are obtained from these principal modules by factorizing their radical, which amounts to kill the Grassmann ``
	www.cpt.univ-mrs.fr /~coque/articles_html/SU2qba/node5.html (634 words)

	IngentaConnect Irreducible representations of Dirac algebra for a constrained sy... (Site not responding. Last check: 2007-10-29)
		IngentaConnect Irreducible representations of Dirac algebra for a constrained sy...
		All possible irreducible representations of the Dirac algebra for a particle constrained to move on a D-dimensional manifold f(x) 0 are explicitly constructed in terms of canonical operators
		It is shown that for D 1 any irreducible representation is uniquely specified by a real parameter belonging to [0, 1), while for D 2 the irreducible representation is unique.
	api.ingentaconnect.com /content/iop/jphysa/2003/00000036/00000023/art00315 (185 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		As a general recipe for questions like this I look up the weight decomposition of the representation in question (did I tell that my background is in Lie algebra/algebraic groups rep theory) and then decompose using the weight decompositions of the irreducible reps of the subalgebra.
		If you tensor two irreducible reps of SU(2), say those of dimensions m and n, corresponding to highest weights (here "weight" is a scalar multiple of "the eigenvalue of the operator L_z") (m-1) and (n-1), you can immediately find an eigenvector corresponding to a maximal weight (m-1)+(n-1)=m+n-2 inside the tensor product.
		No doubt you have seen similar things done with representations of SU(3) (like 3d-rep tensored with its conjugate decomposes into 8+1), where the weight lattice (combinations of eigenvalues of diagonal matrices) is now a two-dimensional pattern (such as a hexagon in the case of octet or a triangle in the case of the 10-d representation).
	www.math.niu.edu /~rusin/papers/known-math/99/Lie_rep (1264 words)

	Re: transformations of irreducible representations
		> assuming that the only only 1-dimensional representation of the lorentz > group is the trivial representation, this would mean that we would > require the 1 component field to be a lorentz scalar.
		Ok, there are a few other, one-dim representations of the (full) Lorentz group which is the signature of the transformation in space (inversion) or time reversal.
		However, it is (I think) not an irreducible representation (that's obvious in the massless case ; I'm less sure in the massive case).
	www.lns.cornell.edu /spr/2003-05/msg0051213.html (288 words)

	The Weyl group action on minuscule weights in Maple. (Site not responding. Last check: 2007-10-29)
		A minuscule representation is an irreducible representation in which all the weights lie in a single Weyl group orbit.
		In fact in some cases we can "see" the irreducibility of the minuscule representation by inspecting the cycle structures of the permutations associated with the Weyl group's elements.
		An invariant subspace of a representation cannot share part of an orbit of weights, so if the cycle structures tell us that there is only one orbit, we must conclude that the representation is irreducible (this assumes that the weight spaces are one-dimensional, which may fail to hold if the highest weight is not minuscule).
	www4.ncsu.edu:8030 /~singer/papers/weyl_permutation.html (438 words)

	PlanetMath: representation theory of $\mathfrak{sl}_2 \mathbb{C}$ (Site not responding. Last check: 2007-10-29)
		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).
		splits into a direct sum of irreducible representations for various non-negative integers as described above.
		This is version 4 of representation theory of
	planetmath.org /encyclopedia/RepresentationTheoryOfMathfraksl_2MathbbC.html (208 words)

	Reducible and irreducible representations (Site not responding. Last check: 2007-10-29)
		We might take two representations and construct from them a new representation, by combining the matrices into larger matrices.
		However, this representation is described as reducible, because it can be split up into smaller representations.
		The irreducible representations are the building blocks for studying the group representations.
	www.astro.cf.ac.uk /undergrad/module/PX3104/tp2/node13.html (92 words)

	Representation Theory (Site not responding. Last check: 2007-10-29)
		Compute the absolutely irreducible representations of the group G over appropriate extension or sub- fields of the given field k.
		The representations returned are inequivalent and are all possible representations, subject to the conditions imposed.
		The "Representations" command returns a list of homomorphisms rho : G to GL(n, K), where K is a field compatible with k.
	www.dtr.isy.liu.se /Magma/text342.html (778 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The Fourier transform of the parent reflections describes a parent structure which is a disordered version of the true structure and is often quite easy to obtain, especially when the inherent disorder is recognised.
		The satellite reflections then define a modulation of the parent structure and possible resulting symmetries can be obtained from the irreducible representations associated with k.
		This situation is associated with the existence of a degenerate irreducible representation.
	www.za.iucr.org /iucr-top/cong/17/iucr/abstracts/abstracts/E0699.html (387 words)

	GAP (repsn) - Chapter 2: Irreducible Representations (Site not responding. Last check: 2007-10-29)
		Let G be a finite group and chi be an ordinary irreducible character of G. In this chapter we introduce some functions to construct a complex representation R of G affording chi.
		The output is a mapping (representation) which assigns to each generator x of G a matrix R(x).
		If chi is an irreducible character of a group G and H is a subgroup of G such that the restriction of chi to H has a linear constituent with multiplicity one, then we call H a character subgroup relative to chi or a chi-subgroup.
	www.gap-system.org /~gap/Manuals/pkg/repsn/html/chap2.html (658 words)

	Representation Theory of Lie Groups - IMS
		We consider the representation of G induced from a character of P. We shall calculate explicitly the action of the Lie algebra of G on this representation.
		Using a cohomological method (etale cohomology) extending that of Deligne and the author (1976), we construct an irreducible representation of G(F_q[[Z]]/(Z^r)) for any "maximal torus" and a generic character of it; for r at least 2, this was stated without proof in a paper I wrote in 1977.
		We study representations of a classical group G which admit certain P-eigendistributions, where P is a parabolic subgroup of G. Through examples, we shall explain how to understand the K-types of G-representations generated by these P-eigendistributions.
	www.ims.nus.edu.sg /Programs/liegroups/abstracts.htm (6923 words)

	Finite and Affine Coxeter Groups
		An affine reflection group is a group generated by reflections in affine space (in other words, real reflections in a hyperplane that does not necessarily pass through the origin).
		A Coxeter group is called affine if it is infinite and it has a representation as a discrete, properly acting, affine reflection group (see [Bou68] for more details on discreteness and proper action).
		A Coxeter group is finite if, and only if, all its irreducible components are finite; a Coxeter group is affine if, and only if all its irreducible components are finite or affine, and at least one component is affine.
	www.math.lsu.edu /magma/text982.htm (1101 words)

	Read about Irreducible (mathematics) at WorldVillage Encyclopedia. Research Irreducible (mathematics) and learn about ... (Site not responding. Last check: 2007-10-29)
		representation theory (group theory), an irreducible representation is a nontrivial
		Similarly, an irreducible module is another name for a
		manifolds, an n-manifold is irreducible if any embedded (n-1) sphere bounds an embedded n-ball.
	encyclopedia.worldvillage.com /s/b/Irreducible_%28mathematics%29 (306 words)

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