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.

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  1. P

    The momentum representation of ##x## and ##[x,p]##

    To deduce the momentum representation of ##[x,p]##, we can see one paradom ##<p|[x,p]|p>=i\hbar## ##<p|[x,p]|p>=<p|xp|p>-<p|px|p>=p<p|x|p>-p<p|x|p>=0## Why? If we deduce the momentum representation of ##x##, we obtain ##<p|x|p>=i\hbar \frac{\partial \delta (p'-p)}{\partial p'}|_{p'=p}##. This...
  2. A

    Finding a Fourier representation of a signal

    Given the following signal, find the Fourier representation, ##V(jf)= \mathfrak{F}\left \{ v(t) \right \}##: ## v(t)=\left\{\begin{matrix} A, & 0\leqslant t\leqslant \frac{T}{3}\\ 2A, & \frac{T}{3}\leqslant t\leqslant T\\ 0, & Else \end{matrix}\right. ## Then sketch ##V(jf)##. Homework...
  3. davidbenari

    Proof check: find adirect product representation for Q8 grou

    Homework Statement Find a direct product representation for the quaternion group. Which are your options? Homework EquationsThe Attempt at a Solution Theorem: The internal direct product of normal subgroups forms a homomorphism of the group...
  4. A

    Representation of conjugate momentum

    We know that in Cartesian position basis the representation of momentum is -ihbar (d/dx) Consider a cylindrical/spherical/whatever curvilinear coordinates. To make life simple, consider a particle constrained to move on a circle so that its position can described by θ only. Suppose we express...
  5. D

    MHB Power Series: Find First 5 Terms of x^2/(1-5x) - Help Needed

    For this function f(x)=x^2/(1-5x). The interval of convergence is (-1/5) < x < (1/5). I tried to differentiate, but got it wrong. Could someone please help?
  6. S

    Calculating Matrix Representation of Linear Function in New Basis

    Homework Statement Let ##f : \mathbb{R}^n \rightarrow \mathbb{R}^m## be a linear function. Suppose that with the standard bases for ##\mathbb{R}^n## and ##\mathbb{R}^m## the function ##f## is represented by the matrix ##A##. Let ##b_1, b_2, \ldots, b_n## be a new set of basis vectors for...
  7. souda64

    Product of two propagators U(-t)U(t) in coord representation

    Here is a mystery I'm trying to understand. Let ##\hat{U}(t) = \exp[-i\hat{H}t]## is an evolution operator (propagator) in atomic units (\hbar=1). I think I'm not crazy assuming that ##\hat{U}(-t)\hat{U}(t)=\hat{I}## (unit operator). Then I would think that the following should hold \left\langle...
  8. B

    Algebra Good book on representation theory of groups

    Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...
  9. M

    Magnetic field representation in Octave

    Homework Statement I need to represent in Octave (using the quiver function) the magnetic field lines around a long wire which carries an electric current around the wire. The magnetic field of an infinitely long straight wire can be obtained by applying Ampere's law.Homework Equations The...
  10. B

    Representation of spin matrices

    I have just started to study quantum mechanics, so I have some doubts. 1) if I consider the base given by the eigenstates of s_z s_z | \pm >=\pm \frac{\hbar}{2} |\pm> the spin operators are represented by the matrices s_x= \frac {\hbar}{2} (|+><-|+|-><+|) s_y= i \frac...
  11. JonnyMaddox

    Hamilton Operator for particle on a circle -- Matrix representation....

    Hey JO. The Hamiltonian is: H= \frac{p_{x}^{2}+p_{y}^{2}}{2m} In quantum Mechanics: \hat{H}=-\frac{\hbar^{2}}{2m}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial x^{2}}) In polar coordinates: \hat{H}=-\frac{\hbar^{2}}{2m}( \frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}...
  12. C

    Geometric representation of a tensor

    Is correct to say that two vectors , three vectors or n vectors as a common point of origin form a tensor ? What is the correct geometric representation of a tensor ? The doubt arises from the fact that in books on the subject , in general there is no geometric representation. Sometimes appears...
  13. naima

    Representation of Lorentz algebra

    i find here a representation of the Lorentz algebra. Starting from the matrix representation (with the ##\lambda## parameter) i see how one gets the matrix form of ##iJ_z## I am less comfortable with the ## -i y\partial_x + x \partial_y## notation Where does it come from? They say that it is a...
  14. K

    Braids as a representation space of SU(5)

    http://arxiv.org/abs/1506.08067 Braids as a representation space of SU(5) Daniel Cartin (Submitted on 23 Jun 2015) The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the...
  15. ognik

    MHB Contour integral representation of Kronecker delta

    I'm rather impressed with complex analysis, but clearly I have a lot to learn. I'm told $ \frac{1}{2\pi i} \oint {z}^{m-n-1} dz $ is a rep. of the kronecker delta function, so I tried to work through that. I used $z = re^{i\theta}$ and got to $ \frac{1}{2\pi}...
  16. Dilatino

    How can I construct the 4D real representation of SU(2)?

    An element of SU(2), such as for example the rotation around the x-axis generated by the first Pauli matrice can be written as U(x) = e^{ixT_1} = \left( \begin{array}{cc} \cos\frac{x}{2} & i\sin\frac{x}{2} \\ i\sin\frac{x}{2} & \cos\frac{x}{2} \\ \end{array} \right) = \left(...
  17. B

    How can a 2D plane and 3D space accurately represent all colors?

    How many dimensions are necessary for repsent ALL colors? 3, 4, more!? What are the better ways for represent all colors inside 2D plane and 3D space. I already tried so much combination, but, a think that never it's 100% good.
  18. askhetan

    Very basic question on Hamiltonian representation?

    I am trying to teach myself DFT (yet again) from books and my maths is only improving at a modest pace to understand how people calculate using QM. So a very basic question now. When a Hamiltonian for a many body system is written as given in page 8 on this presentation...
  19. ChrisVer

    How can an SU(2) triplet be represented as a 2x2 matrix in the Lagrangian?

    Sorry for this "stupid" question... but I am having some problem in understanding how can someone start from let's say an SU(2) triplet and arrive in a 2x2 matrix representation of it in the Lagrangian... An example is the Higgs-triplet models...I think this happens with the W-gauge bosons too...
  20. agent1594

    What type of floating point representation is this?

    In some lecture hand-outs I found the following, In IEEE 754, we just put the binaries of negative fractions in the mantissa without converting to 2C, aren't we? If then, what is the above standard of FP representation? Thanks.
  21. G

    A question about covariant representation of a vector

    Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...
  22. terra

    2j+1 d representation for Poincaré group

    I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...
  23. S

    Direct product representation of a function?

    When do functions have representations as a "direct product"? For example, If I have a function f(x) given by the ordered pairs: \{(1,6),(2,4),(3,5),(4,2),(5,3),(6,1) \} We could (arbitrarily) declare that integers in certain sets have certain "properties": \{ 1,3\} have property A...
  24. A. Neumaier

    When are isomorphic Hilbert spaces physically different?

    In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this...
  25. P

    Standard representation for arbitrary size/precision numbers

    Is there a standard way of representing numbers of arbitrary size or precision for storage in a text file, JSON message, variable etc.? I am thinking of representing integers as decimal strings e.g. "-12345678901234567" and floats as an ordered pair (array) of strings representing decimal...
  26. S

    Superposition & Mixture : Preparation and Representation of

    I have been reading this explanation about superpositions and mixtures. The author takes the example of two non-overlapping regions in space, each covered by a gaussian wavefunction. He goes on to compare the superposition and the mixture made up of those two gaussian functions, based on their...
  27. H

    Linear Transformations and matrix representation

    Assume the mapping T: P2 -> P2 defined by: T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 is linear.Find the matrix representation of T relative to the basis B = {1,t,t2} My book says to first compute the images of the basis vector. This is the point where I'm stuck at because I'm not...
  28. U

    Graphical representation of elevation data.

    What other ways other than a typical contour map and a line graph, are there to graphically present elevation data from a levelling run etc? Any help would be great appreciated. Thank you.
  29. L

    Holstein Primakoff representation

    Holstein Primakoff representation in textbooks is defined by: \hat{S}^+_m=\sqrt{2S}\sqrt{1-\frac{\hat{B}_m^+\hat{B}_m}{2S}}\hat{B}_m \hat{S}_m^-=(\hat{S}^+_m)^+ \hat{S}_m^z=S-\hat{B}_m^+\hat{B}_m And in practical cases it is often to use binomial series for square root, and condition for that is...
  30. A

    Greens function path integral representation

    In my book the path integral representation of the green's function is given as that on the attached picture. But how do you go from the usual trace formula for the Green's function 2.6 to this equation?
  31. I

    Scalar as one dimensional representation of SO(3)

    Hi to all the readers of the forum. I cannot figure out the following thing. I know that a representation of a group G on a vector spaceV s a homomorphism from G to GL(V). I know that a scalar (in Galileian Physics) is something that is invariant under rotation. How can I reconcile this...
  32. U

    Finding parametric representation of a surface

    Homework Statement I am trying to find parametric representation of the right surface of a sphere which was cut along the line y=5. x^2 + y^2 + z^2 = 36 Homework EquationsThe Attempt at a Solution x^2 + y^2 + z^2 = 36 This is an equation of a sphere with radius given by: r^2 = 36 r=6...
  33. T

    MHB Matrix representation on d dimensional vector space

    Hi, i have been going through some elementray reading on group theory. if g(θ) is a group element parameterized by the continuous variable θ. g(θ i )| θ i =identity, if D(θ i ) is a matrix representation on a d dimensional vector space V . What is D(θ i )| θ i =0 ?
  34. c3po

    Find matrix representation for rotating/reflecting hexagon

    Homework Statement Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...
  35. JonnyMaddox

    Tensor product and representations

    Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...
  36. G

    Vector representation of a Quantum State

    In Griffith books of introduction to QM, it say that Quantum State, mathematically represented as a vector. My problem is with understanding what are the components of such a vector. Do I understand it correctly, that, say, in case of a particle in a box, Quantum State, as a vector is a...
  37. S

    Components of adjoint representations

    In the way of defining the adjoint representation, \mathrm{ad}_XY=[X,Y], where X,Y are elements of a Lie algebra, how to determine the components of its representation, which equals to the structure constant?
  38. R

    Quantum Mechanics: Matrix Representation

    Homework Statement What is the matrix representation of ##\mathbb{\hat J}_z## using the states ##|+y\rangle## and ##|-y\rangle## as a basis? Homework Equations ##|\pm y\rangle =\frac{1}{\sqrt{2}}|+z\rangle \pm \frac{i}{\sqrt{2}}|-z\rangle##The Attempt at a Solution A solution was given...
  39. R

    Phasor representation for a mass and spring

    Hi, While writing the Fourier representation for the spring-mass system, we have the equation: (-jωm+B-S/jω).U(jω)=F(jω) with F,U,m,B and S being the force, velocity,mass, damping coefficient and spring constant respectively. From this equation, is it right to infer that : 1) Force lags the...
  40. L

    Complex Representations: Real vs. Complex Lie Algebras

    When do we call a representation complex? What are examples of complex representations? Also, when we say real and complex forms of Lie algebras, is that related to real and complex representation classification? I read that spinors are complex representations of SO(3), because their...
  41. M

    More questions about Lorentz representation

    Hello, First of all, sorry if the question has been asked. I tryied to find some answers but my ignorance goes too deep for any of the previous topics I could find. I'm completely lost when it comes to the Lorentz/Pointcarré groups representations. The part that I don't understand is the...
  42. N

    The double line notation and the adjoint representation

    hi! in the first page of the attached pdf, after the title " 't hooft double line notation", he says that we have to consider the gluon as NxN traceless hermitian matrices to convince ourselves about the double line notation. there is my question: if you want the indices a,b to run from 1 to...
  43. G

    Irreducible representation of GL(D)

    Hi, I'm reading: "Let v_{a} represent a generic element of R^{D}. The action of a non-singular linear operator on this space gives a D-dimensional irreducible representation V of GL(D); indeed, this representation defines the group itself". I have a couple of questions: 1. How do I know that...
  44. P

    Matrix representation of an operator in a new basis

    Homework Statement Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis. Homework EquationsThe Attempt at a Solution...
  45. Fantini

    MHB Momentum operator in the position representation

    Hi! :) I'm trying to understand the following calculation. The book Quantum Mechanics by Nouredine Zettili wants to determine the form of the momentum operator $\widehat{\vec{P}}$ in the position representation. To do so he calculates as follows: $$\begin{aligned} \langle \vec{r} |...
  46. nomadreid

    Continuous set of eigenvalues in matrix representation?

    Let's see if I have this straight: Observables are represented by Hermitian operators, which can be, for some appropriate base, represented in matrix form with the eigenvalues forming the diagonal. Sounds nice until I consider observables with continuous spectra. How do you get something like...
  47. Clear Mind

    Are these quantum states equivalent in Hilbert space representation?

    I've started few days ago to study quantum physics, and there's a thing which isn't clear to me. I know that a quantum state is represented by a ray in a Hilbert space (so that ##k \left| X \right\rangle## is the same state of ##\left| X \right\rangle##). Suppose now to have these three states...
  48. Breo

    Spinorial Field Representation

    What does mean the next (why we write it like this, why is a sum, why first a 0 and secondly a 1/2 and viceversa): $$ (\frac{1}{2} , 0 ) \oplus (0, \frac{1}{2}) $$ ?
  49. F

    Finding Rotational matrix from axis-angle representation

    Homework Statement Given an axis vector u=（-1, -1, -1) , find the rotational matrix R corresponding to an angle of pi/6 using the right hand rule. Then find R(x), where x = (1,0,-1) Homework Equations I found the relevant equation on wikipedia (see attachment) The Attempt at a Solution I feel...
  50. ChrisVer

    3 Dimensional Representation of D3

    I was wondering how can I obtain the three dimensional representation of the Dihedral group of order 6, D_3. If this group has the elements: D_3 = \left \{ e,c,c^2,b,bc,bc^2 \right \} Where c corresponds to rotation by 120^o on the xy plane (so about z-axis) and b to reflections of the x...
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