What is Representation: Definition and 764 Discussions

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.Representation theory is pervasive across fields of mathematics for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:

illuminates and generalizes Fourier analysis via harmonic analysis,
is connected to geometry via invariant theory and the Erlangen program,
has an impact in number theory via automorphic forms and the Langlands program.Second, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

View More On Wikipedia.org
Recent contents Top users
  1. Breo

    A question about wormholes and their typical representation

    Why the wormholes are typically represented as follows: instead: Is the same? in that case why there are two type of draws? Another question, why the wormhole has length? or at least that seems in the draws and in the movies like interstellar or contact.
  2. A

    Proof that the adjoint representation is an endomorphism

    Homework Statement My textbooks takes for granted that, given a Lie group ##g## and its algebra ##\mathfrak{g}##, we have that ##AXA^{-1} \in \mathfrak{g}##. Homework Equations For ##Y## to be in ##\mathfrak{g}## means that ##e^{tY} \in G## for each ##t \in \mathbf{R}## The Attempt at a...
  3. N

    Finding a State-Space Representation for Helicopter Pitch Angle Control

    I need help finding a state-space representation for the given system and locally linearizing it... Objective is to control the pitch angle theta, as an output, of the helicopter by regulating the rotor angle u, as an input. This is what I've done, but apparently it isn't correct, and I...
  4. L

    Sinc^2 as a delta function representation?

    Hi, it's actually not homework but a part of my research. I intuitively see that: \lim_{t \rightarrow \infty} \frac{sin^2[(x-a)t]}{(x-a)^2} \propto \delta(x-a) I know it's certainly true of sinc, but I couldn't find any information about sinc^2. Could someone give me a hint on how I could...
  5. Z

    How to Convert Mechanical System Equations to State Space Form?

    JL*QL'' + BL*QL' + k(QL - Qm) = 0 Jm*Qm'' + Bm*Qm' - k(QL - Qm) = u This is the equation set I have for a motor with a load. QL'' means second derivative and QL' means first derivative. I need to be able to obtain the state space representation of this model where X = [QL;QL';Qm;Qm'] (This is...
  6. C

    Work-Energy Theorem Algebraic Representation

    Homework Statement A car of mass m accelerates from speed v1 to speed v2 while going up a slope that makes an angle θ with the horizontal. The coefficient of static friction is μs, and the acceleration due to gravity is g. Find the total work W done on the car by the external forces. Homework...
  7. maverick280857

    The adjoint representation of a semisimple Lie algebra is completely reducible

    Hi, I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible. Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that...
  8. F

    MHB Help with Maclaurin series representation

    Hello! So, I'm having a bit of a problem with an exercise in my Calculus book. I'm supposed to find the Maclaurin series representation of \frac{1+x^3}{1+x^2} and then express it as a sum. Am I really supposed to differentiate this expression a bunch of times..? That will be very...
  9. K

    SU(3) defining representation (3) decomposition under SU(2) x U(1) subgroup.

    I have been reading Georgi "Lie Algebras in Particle Physics" and on page 183 he mentions how that the SU(3) defining representation decomposes into an SU(2) doublet with hyperchage (1/3) and singlet with hypercharge (-2/3). I am confused on how he knows this. I apologize if this is not the...
  10. J

    Information Representation in Neurons

    Hello everybody, So I've been studying how the brain represents and encodes information. There is ample evidence/info showing that neurons adjust their firing rates and strengths of their synapses in order to encode information and form accessible neural pathways. However I am having trouble...
  11. K

    Exploring Quantum Mechanics: Bra-Ket Representation & Completeness Relation

    I'm new to the concepts of quanum mechanics and the bra-ket representation in general. I've seen in the textbook that the compleatness relation is used all the time when working with the bra and kets. I'm a bit confused about how this relation is being used when applied more than once in a...
  12. K

    MHB Find matrix representation with respect to the basis

    The set of all solutions of the differential equation \d{^2{y}}{{x}^2}+y=0 is a real vector space V=\left\{f:R\to R \mid f^{\prime\prime}+f=0\right\} show that \left\{{e}_{1},{e}_{2}\right\} is a basis for $V$, where {e}_{1}:R \to R, \space x \to \sin(x) {e}_{2}:R \to R, \space x \to...
  13. F

    Quantum states and representation freedom

    Hello Forum, When a system is in a particular state, indicated by a |A>, we can use any basis of eigenvectors to represent it. Every operator that represents an observable has a set of eigenstates. I bet there are operators with only one eigenstate or no eigenstates. There are operators, like...
  14. H

    Alternate Representation of Function

    Hi all, the function that I'm posting about is a piecewise function defined as follows: $$ \Delta(x) = \left\{ \begin{array}{ll} 1 & \quad x = 0 \\ 0 & \quad x \neq 0 \end{array} \right. $$ I decided to call it capital delta because of...
  15. H

    How to select bases for Matrix representation of a point group?

    To represent operations of a point group by matrix we need to choose basis for this representation. What is the criteria for doing that? How to realize that how many bases are necessary for a matrix representation and how to select them? Or could you please give me an elementary reference to...
  16. M

    Hermite representation for integrals?

    Suppose I want an expectation value of a harmonic oscillator wavefunction, then in what way will I write the Hermite polynomial of nth degree into the integral? I have a link of the representation, but don't know what to do with them? http://dlmf.nist.gov/18.3
  17. gfd43tg

    Understanding IEEE Representation for Single Precision

    Hello, For the IEEE representation of a number, I wanted to ask something for clarification. For single precision, you have 3 parts: S, Exponent, and Fraction. The S takes 1 bit (1 slot) Exponent is 8 bits (8 slots) Fraction is 23 bits (23 slots). I was watching a video and it helped me...
  18. gfd43tg

    Every other odd number representation

    Hello, Is there any formula that describes every other odd number, for ##n = 1,2,\dots##? I can't seem to find anything that does it on the web. Something that would do 1,5,9,13,...
  19. C

    Adjoint representation vs the adjoint of a matrix

    In my limited study of abstract Lie groups, I have come across the adjoint representation ##Ad: G \to GL(\mathfrak{g})## on the lie algebra ##\matfrak{g}##. It is defined through the conjugation map ##C_g(h) = ghg^{-1}## as the pushforward ##C_{g*}|_{g=e}: \mathfrak{g} \to \mathfrak{g}##...
  20. C

    Group Representation: Understanding SO(3), SU(2), and the Clebsch-Gordan Theorem

    Good morning I'me french so excuse my bad language : so in this course : http://lapth.cnrs.fr/pg-nomin/salati/TQC_UJF_13.pdf take a look at page 16. They say that all rotation auround a unitary vector \vec{u} of angle \theta in the conventionnal space could be right like this with the matrix...
  21. Greg Bernhardt

    What is a Group Representation and How Does it Act on a Vector Space?

    Definition/Summary A group representation is a realization of a group in the form of a set of matrices over some algebraic field, usually the complex numbers. A representation is irreducible if the only sort of matrix that commutes with all its matrices is a sort that is proportional to...
  22. BruceW

    Are Faithful Representations the Most Interesting Group Representations?

    Hello everyone! I've been learning some basic group theory (I'm new to the subject). And I had a (hopefully) fairly simple question. OK, so a 'faithful representation' is defined as an injective homomorphism from some group G to the Automorphism group of some object. Let's call the object S for...
  23. stevendaryl

    Is Choice of Spinor Representation a Gauge Symmetry?

    In the Dirac equation, the only thing about the gamma matrices that is "fixed" is the anticommutation rule: \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} We can get an equivalent equation by taking a unitary matrix U and defining new spinors and gamma-matrices via...
  24. D

    Prove an integral representation of the zero-order Bessel function

    Homework Statement In section 7.15 of this book: Milonni, P. W. and J. H. Eberly (2010). Laser Physics. there is an equation (7.15.9) which is an integral representation of the zero-order Bessel function: J_0(\alpha\rho)=\frac{1}{2\pi}\int^{2\pi}_{0}e^{i[\alpha(xcos{\phi}+ysin{\phi})]}d\phi...
  25. Q

    Group, Symmetries and Representation

    I'm starting to learn about particle physics but I really want to see the whole picture before going deep. Here is what I know: - There are symmetries in quantum physics, which are symmetry operators commute with the Hamiltonian (translation operators, rotation operators...) which act on a...
  26. Seydlitz

    Notation used in matrix representation of linear transformation

    Hello guys, Let ##T: \mathbb{R^2} \to \mathbb{R^2}##. Suppose I have standard basis ##B = \{u_1, u_2\}## and another basis ##B^{\prime} = \{v_1, v_2\}## The linear transformation is described say as such ##T(v_1) = v_1 + v_2, T(v_2) = v_1## If I want to write the matrix representing ##T##...
  27. J

    Representation symmetric, antisymmetric or mixed

    Hi, While studying Lie groups and in particular the representations I have some trouble with determining whether a representation is a symmetric, antisymmetric or mixed tensor product of the fundamental representations. I'm working with SO(5) to get an understanding about the actual...
  28. M

    Taylor series representation help

    Homework Statement Find the Taylor Series of x^(1/2) at a=1 Homework Equations i have no idea how to do the representation, i believe our professor does not want us to use any binomial coefficients The Attempt at a Solution i got the expansion and here's my attempt at the...
  29. Xenosum

    SU(2) Spin-1/2 Representation Question

    In Howard Georgi's Lie Algebras in Particle Physics (and other texts I'm sure), it is determined that the Pauli matrices, \sigma_1 \sigma_2 and \sigma_3, in 2 dimensions form an irreducible representation of the SU(2) algebra. This is a bit confusion to me. The SU(2) algebra is given by...
  30. W

    Adjointness and Basis Representation

    Hi, Let V be a fin. dim. vector space over Reals or Complexes and let L: V-->V be a linear operator. I am just curious about how to use a choice of basis for general V, to decide whether L is self-adjoint. The issue, specifically, is that the relation ## L= L^T ## ( abusing notation ; here L...
  31. P

    Representation of two relation matrices

    Homework Statement The Attempt at a Solution I know what relations are individually but what do I do to represent the composition of both? Is it some matrix operation? Would I multiply them, but instead of adding I use the boolean sum?
  32. S

    Is this representation of a photon correct

    http://http://education-portal.com/cimages/multimages/16/photonpar.jpg is this image correct does the photon move in a pattern like a wave?
  33. L

    What is the use of infinite-dimensional representation of group

    What is the use of infinite-dimensional representation of lie group? Now, I know Hilbert space is infinite-dimensional, and physical states must be in Hilbert space. However, for massive fields, the transformation group is SO(3), its unitary representation is finite. For massless fields, the...
  34. V

    Matrix form of Lie algebra highest weight representation

    I am currently working with Lie algebras and my research requires me to have matrix representations for any given Lie algebra and highest weight. I solved this problem with a program for cases where all weights in a representation have multiplicity 1 by finding how E_\alpha acts on each node of...
  35. L

    Fourier transform. Impulse representation.

    ##\varphi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int^{\infty}_{-\infty}dx\psi(x)e^{-\frac{ipx}{\hbar}}##. This ##\hbar## looks strange here for me. Does it holds identity ##\int^{\infty}_{-\infty}|\varphi(p)|^2dp=\int^{\infty}_{-\infty}|\psi(x)|^2dx=1##? I'm don't think so because this ##\hbar##. So...
  36. K

    Question on the 2-dim representation of the Lorentz group

    Hello! I'm currently reading some QFT and have passed the concept of Weyl spinors 2-4 times but this time it didn't make that much sense.. We can identify the Lorentz algebra as two su(2)'s. Hence from QM I'm convinced that the representation of the Lorentz algebra can be of dimension (2s_1 +...
  37. J

    Ideal representation for vectors/covectors

    A vector can be expressed by a column matrix or by a row matrix, however is preferably use the column matrix for represent vectors and row matrix for covectors. So, analogously, we can express a vector v as v1 e1 + v2 e2 or as v1 e1 + v2 e2. But I think that a vector is represented more...
  38. R

    Find arcsine(-2) using the rectangular representation of sin w

    Homework Statement Find \sin^{-1}(-2) by writing [tex]sin w = -2[/itex] and using the rectangular representation of \sin w Homework Equations Rectangular representation of \sin w The Attempt at a Solution I think my biggest problem here is I have literally no idea what the...
  39. K

    Representation of SUSY Algebra

    I have some questions about representations of SUSY algebra. (1) Take ##N=1## as an example. Massive supermultiplet can be constructed in this way: $$|\Omega>\\ Q_1^\dagger|\Omega>, Q_2^\dagger|\Omega>\\ Q_1^\dagger Q_2^\dagger|\Omega>$$ I understand the z-components ##s_z## of the last...
  40. M

    Scalar in adjoint representation

    Hello, people. I'm studying (as an exercise) the breaking of an SU(3) gauge group to SU(2) x U(1) via a Higgs mechanism. The scalar responsible for the breaking is \Phi, who transforms under the adjoint representation of SU(3) (an octet). First of all I want to construct the most general...
  41. C

    Representation of e in terms of primes

    We can represent π, in terms of primes by using Euler's product form of Riemann Zeta. For example ζ(2)=(π^2)/6= ∏ p^2/(p^2-1). Likewise, is there a representation of e that is obtained by using only prime numbers?
  42. A

    Understanding field representation of force

    I am reading the book "The Evolution of Physics". I have a doubt in the topic "The field as representation". In this topic authors give the example of gravitational force represented as a field. In the following image the small circle represents an attracting body(say sun) and the lines are the...
  43. O

    How can I rewrite the series to apply the formula without changing the result?

    \sum\limits_{m=-N}^N e^{-i m c} = \frac{sin[0.5(2N+1) c]}{sin[0.5 c]} I have to show the equality. But I'm absolutely dumbfounded how to even begin. I always hated series. I tried to use Euler's identity. e^{-i m c} = cos(mc) - i sin(mc) Then I tried to sum over the 2 terms separately...
  44. K

    Is scalar in adjoint representation always real

    This is a short question. I don't know why, but somehow I have the impression that scalar in adjoint representation should be real. Now I highly doubt this statement, but I have no idea how to disprove it. Can anyone give me a clear no? Thanks,
  45. alyafey22

    MHB Limit representation of Euler-Mascheroni constant

    We have the following functional equation of digamma \psi(x+1)-\psi(x)=\frac{1}{x} It is then readily seen that -\gamma= \lim_{z\to 0} \left\{ \psi(z) +\frac{1}{z} \right\} Prove the following -\gamma = \lim_{z \to 0} \left\{ \Gamma(z) -\frac{1}{z} \right\}
  46. L

    Group S3: Irreducible vs Reducible Representation

    Homework Statement ##e = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}##, ##a =\frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}##. ##b...
  47. L

    Irreducible representation. I'm confused.

    All commutative groups have one dimensional representation ##D(g_i)=1, \forall i## I understand what is representation. Also I know what is one, two... dimensional representation. But what is irreducible representation I do not understand. How you could have two dimensional irreducible...
  48. L

    Why Is a Group with Identical Elements Considered Unfaithful?

    If I have some group representation ##D(e)=1##, ##D(s)=1## where ##e\neq s## it is called unfaithfull because it is not isomorphism. If I denote this group by ##(\{1,1\},\cdot)##. My question is how I treat this set as a two element one, when I have only one element in the set? I'm a bit...
  49. L

    Showing the Representation of the Delta Function

    Homework Statement Show that ##\frac{1}{\pi}\lim_{\epsilon \to 0^+}\frac{\epsilon}{\epsilon^2+k^2}## is representation of delta function.Homework Equations ##\delta(x)=\frac{1}{2 \pi}\int^{\infty}_{-\infty}dke^{ikx}## The Attempt at a Solution...
  50. K

    The representation of Lorentz group

    The lorentz group SO(3,1) is isomorphic to SU(2)*SU(2). Then we can use two numbers (m,n) to indicate the representation corresponding to the two SU(2) groups. I understand (0,0) is lorentz scalar, (1/2,0) or (0,1/2) is weyl spinor. What about (1/2, 1/2)? I don't get why it corresponds to...
Back
Top