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Topic: Modular representation

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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)

	Modular representation theory - Wikipedia, the free encyclopedia
		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.
		Modular representation theory was developed by Richard Brauer from about 1940 onwards to provide more detailed information linked to the structure of G.
	en.wikipedia.org /wiki/Modular_representation_theory (386 words)

	Modular form (Site not responding. Last check: 2007-10-15)
		A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition.
		Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves.
		Hilbert modular forms are functions in n variables, each a complex number in the upper half plane, satisfying a modular relation for 2×2 matrices with entries in a totally real number field.
	www.portaljuice.com /modular_form.html (1121 words)

	Pure Mathematics Research, Department of Mathematics, Univ. of Manchester, UK
		In modular representation theory of Lie algebras, irreducible representations of the $p$-Lie algebra of a simple algebraic group, having a subregular nilpotent $p$-character, are studied.
		Modular representation theory of Lie algebras is now a very active and attractive field due to deep interactions with representation theory of quantum groups at roots of unity and some very recent discoveries such as Premet's proof of the Kac-Weisfeiler conjecture and Jantzen's work on subregular representations of $sl(n)$ and $so(2n+1)$.
		In modular representation theory of $S_n$ and $GL(n)$, the differential operator algebra and the Steenrod algebra are used to study the natural action of the above two groups on the polynomial ring in $n$ variables.
	www.ma.man.ac.uk /DeptWeb/Groups/Pure/junk.txt (512 words)

	Genome Biology \| Full text \| Modular decomposition of protein-protein interaction networks
		Modular decomposition derives the logical rules of how to combine proteins into the actual functional complexes by identifying groups of proteins acting as a single unit (sub-complexes) and those that can be alternatively exchanged in a set of similar complexes.
		The nodes of the modular decomposition are labeled in three ways (Figure 2): as series when the direct descendants are all neighbors of each other; as parallel when the direct descendants are all non-neighbors of each other; and by the structure of the module otherwise (the so-called prime module case).
		Modular decomposition groups together proteins with a similar function: the catalytic subunits Pph21 and Pph22 as alternatives in a parallel module and the regulatory subunits Cdc55 and Rts1 in another parallel module.
	www.genomebiology.com /2004/5/8/R57 (5439 words)

	Module - Wikipedia, the free encyclopedia
		A module is a self-contained component of a system, which has a well-defined interface to the other components; something is modular if it is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair of the components.
		In mathematics, there are a number of unrelated concepts which use the words module or modular.
		For hypothesized modules in mental processes, see modularity of mind.
	en.wikipedia.org /wiki/modular (185 words)

	Citation for James Alexander Green (Site not responding. Last check: 2007-10-15)
		Green was probably the first to realise the power of studying representation theory over complete discrete valuation rings and this work, together with that of two other papers in that period, provided the impetus for focussing attention on modules, in contrast to Brauer’s original character theoretic approach.
		More recently, he has made substantial contributions to the study of representations of quantum groups via a relationship with the Hall algebras that he had studied earlier in his 1955 paper.
		Green has written key papers on many other topics, including the converse to Brauer's induction theorem and modular representations of finite groups of Lie type, and he introduced the concepts of Mackey functors, the Green ring, and G-algebras.
	www.lms.ac.uk /activities/prizes_com/citations01/green.html (386 words)

	The emergence of cooperative playing routines: optimality and learning (Site not responding. Last check: 2007-10-15)
		The modular representation (in terms of flags) of level II states has several advantages: the most important of these is that it captures the essential aspects of the game dynamics, according to the structural graph mentioned before.
		Moreover the universality of the modular representation leads to a finer and more reliable statistics of the experimental data and is also generalizable to complex games with more than two levels.
		When written in the modular representation the number of possible distinct hands of level II reduces to 60: given the card in target, whose role is to define the card-values of the various flags, there remain 5 positions among which to distribute 3 flags, and thus
	www-ceel.gelso.unitn.it /papers/paolo/index.html (3459 words)

	sci.math FAQ: Wiles attack (Site not responding. Last check: 2007-10-15)
		Start with a mod p representation of the Galois group of Q which is known to be modular.
		One of these is related to the Selmer group of the symmetric square of the given modular lifting of rho_p, and the other is related (by work of Hida) to an L -value.
		Let X denote the modular curve whose points correspond to pairs (A, C) where A is an elliptic curve and C is a subgroup of A isomorphic to the group scheme E[5].
	www.faqs.org /faqs/sci-math-faq/FLT/Wiles (1502 words)

	Introduction
		A representation of another class of modular lattices was given by Baer [1942] when he developed a unified theory of projective spaces and finite Abelian groups.
		Further systematic research on the algebraic representation of modular lattices containing a homogeneous basis by free modules was done by Artmann [1968] and by Day and Pickering [1983].
		This is the generalization of the fundamental theorem of projective geometry* and hence, the representation of mappings between submodule lattices.
	www.elsevier.com /homepage/saj/504595/21a.htm (1128 words)

	Eigenfaces/Photobook Demo (Site not responding. Last check: 2007-10-15)
		This can be viewed as either a modular or layered representation of a face, where a coarse (low-resolution) description of the whole head is augmented by additional (higher-resolution) details in terms of salient facial features.
		The reconstruction error (or residual) of the principal component representation (referred to as the distance-from-face-space) is a an effective indicator of a match.
		What is surprising is that (for this small dataset at least) the eigenfeatures alone were sufficient in achieving an (asymptotic) recognition rate of 95% (equal to that of the eigenfaces).
	www-white.media.mit.edu /vismod/demos/facerec/basic.html (1408 words)

	Vignettes on automorphic and modular forms, representations, L-functions, and number theory (Site not responding. Last check: 2007-10-15)
		We want to prove that the singular homology of quotients X/Gamma is the group homology of Gamma, under some mild conditions on X (such as that X be a ball).
		Banach space representations of real reductive groups are of moderate growth.
		Standard basic features of representation theory of p-adic reductive groups: exactness of Jacquet module functors, Jacquet's lemmas, admissibility and finite-generation of Jacquet modules of admissible finitely-generated smooth representations.
	www.math.umn.edu /~garrett/m/v (1093 words)

	The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles (Site not responding. Last check: 2007-10-15)
		An eigenform with rational eigenvalues yields a homomorphism $\widehat{\bold{T}}\rightarrow\bold{Z}_l$, and $\rho$ induces the $l$-adic representation that is given by the associated elliptic curve $E$, describing the Galois action on all $l^n$-division points of $E$.
		This is known to be congruent to a modular representation, and then the universal lifting of this representation is proven modular, which is the core of the proof.
		According to the theorem of Ribet and others (used for $l=3$ and not for $l$ the exponent of Fermat's equation), the Galois representation belonging to the curve modulo 3 is modular of level 3.
	www.geocities.com /Vienna/Strasse/7684/fermat.html (2071 words)

	Modular Utility Representation (Site not responding. Last check: 2007-10-15)
		Decision-theoretic approaches require that representations of objectives possess a firm semantics in terms of utility functions, yet provide the flexible compositionality needed for practical preference modeling for planning systems.
		Modularity, or separability in specification, is the key representational feature enabling this flexibility.
		In the context of utility specification, modularity corresponds exactly to well-known independence concepts from multiattribute utility theory, and leads directly to approaches for composing separate preference specifications.
	ai.eecs.umich.edu /people/wellman/pubs/aips92.html (125 words)

	Abstract (Site not responding. Last check: 2007-10-15)
		The first construction uses the modular groups and shows how one can get from it a representation of the Poincar\'e group which acts as symmetry group on the net of local algebras.
		Thereafter, the use of the modular conjugation for the construction of vacuum representations will be discussed.
		Finally the concept of modular localization, introduced by Brunetti, Guido and Longo, will be discussed together with some consequences for Wigners infinite spin representations for massless particles.
	www.lqp.uni-goettingen.de /papers/04/03/04030300.html (202 words)

	Representation Theory Of Symmetric Groups (ResearchIndex) (Site not responding. Last check: 2007-10-15)
		Abstract: this article we will give an overview of the new Lie theoretic approach to the p-modular representation theory of the symmetric groups and their double covers that has emerged in the last few years.
		There are in fact two parallel theories here: one for the symmetric groups S n involving the ane Kac-Moody algebra of type A p 1, and one for their double covers b S n involving the twisted algebra of type A p 1.
		Representations Of The Symmetric Group Are Reducible Over..
	citeseer.ist.psu.edu /626921.html (612 words)

	Pure Mathematics Research, Department of Mathematics, Univ. of Manchester, UK
		In modular representation theory of Lie algebras, irreducible representations of the p-Lie algebra of a simple algebraic group, having a nilpotent p-character, are studied.
		It is established that a similar relationship exists between the algebras arising in subregular representation theory and noncommutative deformations of Klein singularities similar to those introduced recently by Hodges, Crawley-Boevey and Holland.
		Modular representation theory of Lie algebras is now a very active and attractive field due to deep interactions with representation theory of quantum groups at roots of unity and some recent discoveries such as Premet's proof of the Kac-Weisfeiler conjecture and Jantzen's work on subregular nilpotent representations.
	www.ma.man.ac.uk /DeptWeb/Groups/Pure/Algebra.html (498 words)

	Chapter 3 - The Minkowski Question Mark and the Modular Group (Site not responding. Last check: 2007-10-15)
		This matrix representation of the generators can be used to manipulate and evaluate continued fractions; indeed, the modular group is known as being one way to conveniently work with continued fractions.
		The dyadic and higher-order interval representations are not symplectic.
		Finally, we note that the interval representation is a topology, and that it is not exactly a trivial topology for a subset of the modular group.
	www.linas.org /math/chap-minkowski/chap-minkowski.html (5943 words)

	The Proof of Fermat's Last Theorem
		Specifically, it cannot be modular in the sense that there exists a modular form which gives rise to the same Galois representation.
		To show that E is modular, we have to show this representation is modular in a suitable sense.
		This is done mainly by working with the theory of representations as much as possible, without specific reference to the curve E. The proof uses a concept called "deformation", which suggests intuitively what goes on in the process of lifting.
	www.mbay.net /~cgd/flt/flt08.htm (1543 words)

	Psycoloquy 5(88): Representation in Modular Networks (Site not responding. Last check: 2007-10-15)
		Two features of the system seem to be particularly important: first, the modular decomposition of the task into a set of simpler tasks, and second, the closely related use of a central lexicon, with representations evolved through Miikkulainen's FGREP technique.
		Although the representations are systematic and compositional, there is a crucial difference between DISCERN and standard symbolic systems in the relationship between the representations and the processes which operate over them.
		Given the use by the system of systematic, compositional representations, it is not clear that DISCERN is not both modular connectionist and hybrid symbolic/connectionist.
	www.cogsci.ecs.soton.ac.uk /cgi/psyc/newpsy?5.88 (2164 words)

	tino lukaschek's current studies (Site not responding. Last check: 2007-10-15)
		general modular representation theory of finite dimensional associative algebras
		associated with my work for the thesis, i want to establish a sound knowledge in modular representation theory of associative algebras.
		most of the classical theory of associative algebras is applied to the modular representation theory of group algebras.
	homepages.uel.ac.uk /T.H.Lukaschek/cust.html (279 words)

	Modular Lie Algebras (Site not responding. Last check: 2007-10-15)
		Roughly speaking, many features from the classical structure and representation theory can be salvaged as long as the Lie algebra to be studied comes from a smooth algebraic group scheme.
		In fact, the representation theory of reductive Lie algebras (i.e., those associated to reductive algebraic groups) is currently a rather active field of research.
		Much of the current work on modular Lie algebras focuses on support varieties and the investigation of irreducible representations of reductive Lie algebras.
	www.mathematik.uni-bielefeld.de /~rolf/ModLie.html (475 words)

	Janko group (Site not responding. Last check: 2007-10-15)
		Janko found a modular representation in terms of 7 × 7 matrices in the field of eleven elements, with generators given by
		It can also be constructed via an underlying geometry, as was done by Weiss, and has a modular representation of dimension eighteen over the finite field of nine elements, which can be expressed in terms of two generators.
		It has a modular representation of dimension 112 over the finite field of two elements, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally.
	www.omniknow.com /common/wiki.php?in=en&term=Hall-Janko-Wales_group (754 words)

	Topics in Modular Representation Theory (L24) (Site not responding. Last check: 2007-10-15)
		This course will consist of some topics in the representation theory of finite groups and algebras.
		One possibility is that I may try and explain the meaning and occurrence of enumerative combinatorics in classical representation theory.
		It turns out that subtle homological properties of the category of representations have combinatorial shadows; these are quite rigid.
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node4.html (204 words)

	University of Michigan Arithmetic Seminar (Site not responding. Last check: 2007-10-15)
		For function fields, a Galois representation is the same as a representation of the fundamental group of a curve X over a finite field.
		The space of functions on the set of rank n vector bundles on X carries an action of a Hecke algebra, and modularity of a representation means there is a corresponding eigenfunction on this set.
		In this context, we construct a map from a completion of the Hecke algebra to the space of deformations of a fixed residual representation and prove it is an isomorphism.
	www.math.lsa.umich.edu /seminars/arithmetic/2004/Dec13.html (164 words)

	Tech Reports: HPL-98-134: Efficient Arithmetic in GF
		Abstract: A representation of the field GF(2 super n) for various values of n is described, where the field elements are palindromic polynomials, and the field operations are polynomial addition and multiplication in the ring of polynomials modulo x (super 2n+1)-1.
		As such, the suggested palindromic representation inherits the advantages of two commonly-used representations of finite fields, namely, the standard (polynomial) representation and the optimal normal basis representation.
		Modular polynomial multiplication is well suited for software implementations, whereas the optimal normal basis representation admits efficient hardware implementations.
	www.hpl.hp.com /techreports/98/HPL-98-134.html (167 words)

	Modularity of Development
		Riedl predicts that the evolution of the genetic representation of phenotypic characters tends to favor those representations which imitate the functional organization of the characters.
		A modular representation of two character complexes C1 and C2 is given if pleiotropic effects of the genes are more frequent among the members of a character complex than among members of different complexes (see Fig.
		The genetic representation is modular because the pleiotropic effects of the genes M1={G1, G2, G3} have primarily pleiotropic effects on the characters in C1 and M2={G4, G5, G6} on the characters in complex C2.
	www.cbc.yale.edu /old/cce/papers/ALife/node4.html (628 words)

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