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. A

    Eigenstate and real space representation

    Hi All, I was going through a paper on quantum simulations. Below is an extract from the paper; I would be obliged if anyone can help me to understand it: We will use eigenstate representation for transverse direction(HT) and real space for longitudinal direction(HL) Hamiltonians. HL=...
  2. V

    Mathematical Representation of a Traveling Wave

    Homework Statement A wave is represented by y = A sin (kx + ωt). Draw two cycles of the wave from x = 0 to x = 2λ at a) t = 0; b) t = T/4, where T = 1/f = 2∏/ω Homework Equations y = A sin (kx+ωt) k = 2∏/λ (number of wave peaks) The Attempt at a Solution I had a really hard...
  3. K

    MHB Permutation representation argument validity

    Hello, I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you. **Problem statement:** Let $G$ be a group of order 150. Let $H$ be a subgroup of $G$ of order 25. Consider the action of $G$ on $G/H$ by left...
  4. K

    MHB Surjectivity for permutation representation of a group action

    I am having trouble proving that my function is surjective. Here is the problem statement: Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the permutation representation of this action, to show that T is isomorphic to a subgroup of $S_4$...
  5. T

    Matrix representation of linear transformation

    Let V and W be two finite-dimensional vector spaces over the field F. Let B be a basis of V, and let C be a basis of W. For any v 2 V write [v]B for the coordinate vector of v with respect to B, and similarly [w]C for w in W. Let T : V -> W be a linear map, and write [T]C B for the matrix...
  6. G

    Power series representation

    Hi, I'm trying to find the series representation of f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt . I have found it ussing the Maclaurin series, differenciating multiple times and finding a pattern. But I think it must be an eassier way, using the power series of elementary functions. I know that...
  7. A

    Is chirality dependent on the representation of the gamma matrices?

    Hi, In QFT we define the projection operators: \begin{equation} P_{\pm} = \frac{1}{2} ( 1 \pm \gamma^5) \end{equation} and define the left- and right-handed parts of the Dirac spinor as: \begin{align} \psi_R & = P_+ \psi \\ \psi_L & = P_- \psi \end{align} I was wondering if the left- and...
  8. S

    Phrase 'the field is in [certain] representation of a [certain] group'

    I have a question about quantum field theory. What does the phrase 'the field is in [certain, e. g. fundamental] representation of a [certain, e. g. SU(2)] group' mean? I know mathematical definitions of groups and their representations, but what does this specific phrase mean?
  9. R

    Matrix Representation of Operators

    Homework Statement This is quite a long problem, and I have most of it figured out, but I am getting stuck on the very last part of the problem. My problem is I do not understand how to find \left\langle\psi|a_1\right\rangle in the very last line. Is...
  10. D

    Show that the natural representation of S3 is a direct sum of irreps

    Homework Statement Hey everyone! So to elaborate the title a bit more: basically I have to show that the natural representation of S_{3} is a direct sum of the one-dimensional irreducible representation and the two-dimensional irreducible representation of S_{3}. Homework Equations Im...
  11. Q

    Open Loop Representation of Closed Loop System

    I remember having studied that closed loop systems can be represented by open loop systems. But that seems weird..if it were possible for both the types of systems to have the same transfer function, why would they behave differently?
  12. F

    A problem of momentum representation

    Given [x,p] = i * h-bar, prove that <p|X|p'> = [i * h-bar / (p' - p)] * δ(p - p'). I don't understand why commutator matters with this proof?
  13. jegues

    Understanding State Space Representation for First-Order Dynamic Systems

    Homework Statement A first-order dynamic system is represented by the differential equation, 5\frac{dx(t)}{dt} + x(t) = u(t). Find the corresponding transfer function and state space reprsentation. Homework Equations The Attempt at a Solution Putting the equation in the...
  14. polygamma

    MHB An integral representation of the Hurwitz zeta function

    For $ \text{Re} (a) >0$ and $\text{Re} (s)>1$, the Hurwitz zeta function is defined as $ \displaystyle \zeta(s,a) = \sum_{n=0}^{\infty} \frac{1}{(a+n)^{s}} $. Notice that $\zeta(s) = \zeta(s,1)$. So the Hurwitz zeta function is a generalization of the Riemann zeta function. And just like the...
  15. V

    Representation of numbers in quantum bits

    Hi, A bit is a fundamental unit of information, classically represented as a 0 or 1 in your digital computer. I now number 100 is written in classical bits 0 and 1 as 1100100.Then How to represent 100 in qbits. cheers!
  16. polygamma

    MHB Another integral representation of the Riemann zeta function

    Here is another integral representation of $\zeta(s)$ that is valid for all complex values of $s$. It's similar to the first one, but a bit harder to derive.$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} +...
  17. A

    Use of phasor representation in physics

    Why do we use phasor representation in physics..For example,why we need maxwells equation in phasor form as well??
  18. L

    Tensor techniques in $3\otimes\bar 3$ representation of su(3)

    Hi everyone! I would like to ask you a very basic question on the decomposition 3\otimes\bar 3=1\oplus 8 of su(3) representation. Suppose I have a tensor that transforms under the 8 representation (the adjoint rep), of the form O^{y}_{x} where upper index belongs to the $\bar 3$ rep and the...
  19. zhangwfjh

    On representation of expansion in fluid flow

    In Helmholtz original thesis On integrals of the hydrodynamical equations, which express vortex-motion, he mentioned in the first section that the change undergone by an arbitrary infinitesimal volume of water under the time dt is composed of three different motions. One of them is an expansion...
  20. polygamma

    MHB An integral representation of the Riemann zeta function

    Show that $\displaystyle \zeta(s) = \frac{2^{s-1}}{1-2^{1-s}} \int_{0}^{\infty} \frac{\cos (s \arctan t)}{(1+t^{2})^{s/2} \cosh \left( \frac{\pi t}{2} \right)} \ dt $The cool thing about this representation is that it is valid for all complex values of $s$ excluding $s=1$. This integral is...
  21. J

    MHB Isomorphic representation

    Consider the complete graph with 5 vertices, denoted by K5. C. Find an isomorphic representation (graph) of K5. Give the isomorphism mappings. Can someone please tell me if this is correct? One dot on graph = K1 One dot on graph = K2 One dot on graph = K3 One dot on graph = K4 One dot on...
  22. A

    Confusion about the definition of adjoint representation and roots.

    Hi, I'm getting a bit confused about the adjoint representation. I learned about Lie algrebras using the book by Howard Georgi (i.e. it is very "physics-like" and we did not distinguish between the abstract approach to group theory and the matrix approach to group theory). He defines the...
  23. K

    Representation of a 1 phase transmission line as line charges

    Hi, According to what I understood, when a terminal is connected to a line, it causes electrons to flow in one direction. And so for a single phase transmission line in parallel if connected to supply, then in one side electrons will be flowing in one direction and the same electrons will be...
  24. J

    Criterion for (non)decomposability of a representation?

    Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma. My question is...
  25. lalo_u

    Representation of SU(2)xU(1)

    Well, i´m trying to understand this: I´ve got a representation of SU(2)_L\otimes U(1)_Y such that the left lepton doublets can be represented as (2, -1) and lepton singlets rights as (1, -2). Then I can be left antiparticles bilinear representations as (2,1)\times(2,1) or...
  26. Y

    Double check the derivation integral representation of Bessel Function

    I am reading the article Mirela Vinerean: http://www.math.kau.se/mirevine/mf2bess.pdf On page 6, I have a question about e^{\frac{x}{2}t} e^{-\frac{x}{2}\frac{1}{t}}=\sum^{\infty}_{n=-\infty}J_n(x)e^{jn\theta}=\sum_{n=0}^{\infty}J_n(x)[e^{jn\theta}+(-1)^ne^{-jn\theta}] I think there is a...
  27. C

    Can Elementary Particles be related with irreducible representation?

    Hi, I am quite naive in Particle Physics, and I have a question that Can Elementary Particles be related with irreducible representation? Could we say scalar, vector, and spinor are irreducible representation of SO(3)? Thanks a lot! I also wish I could have some reference on...
  28. M

    Why is (1/2,1/2) the vector representation?

    Hello, I'm reading Zee's book 'Quantum Field Theory in a Nutshell', the chapter about Lorentz group representations at the moment. In the end of the chapter there is suggested an exercise - "Show by explicit computation that (1/2,1/2) is indeed the Lorentz vector". And I just can't figure it...
  29. H

    General matrix representation of lorentz boost

    Hello! I'm trying to derive the general matrix form of a lorentz boost by using the generators of rotations and boosts: I already managed to get the matrices that represent boosts in the direction of one axis, but when trying to combine them to get a boost in an arbitrary direction I always...
  30. lonewolf219

    How to find 3D representation of SU(2)

    Homework Statement Find the 3D representation of what I think are the commutators [T_a,T_b] for the SU(2) groupHomework Equations I think the generators(X_i) in SU(2) group are the 3 Pauli matrices, which are 2X2 matrices... I assume I need to find the matrices for these generators as 3x3...
  31. Y

    Representation equivalent to a unitary one

    Hey guys, How come a representation \rho of a group G is always equivalent to a unitary representation of G on some (inner product) space V ? Can anyone provide a good source (book, preferably) which states a proof? Thanks
  32. Y

    Reducibility tensor product representation

    Hello everyone, Say I have two irreducible representations \rho and \pi of a group G on vector spaces V and W. Then I construct a tensor product representation \rho \otimes \pi : G\to \mathrm{GL}\left(V_1 \otimes V_2\right) by \left[\rho \otimes \pi \right] (g) v\otimes w = \rho (g) v...
  33. Y

    Direct sum and product representation

    Hi everyone, I'm having some trouble with the concept of the direct sum and product of representations. Say I have two representations \rho_1 , \rho_2 of a group G on vector spaces V_1, V_2 respectively. Then I know their direct sum and their product are defined as \rho_1 \oplus \rho_2 : G...
  34. V

    Representation of linear operator using series ?

    representation of linear operator using "series"? I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by: $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$ I'm not sure if that's...
  35. Y

    Is the wigner D function a representation of SO(3)?

    Hello everyone, I'm reading a bit about the Wigner D matrix, defined by \mathscr{D}\left(\hat{n},\phi \right) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \hat{n}]. Now I'm wondering : is the map \pi : \text{SO(3)} \to \text{GL}\left( \mathscr{H} \right) given by R\left(\hat{n},\phi...
  36. Fernando Revilla

    MHB Johnathan's question at Yahoo Answers (Power series representation)

    Here is the question: Here is a link to the question: Help with this power series representation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  37. J

    Surfaces without an explicit representation.

    Ok this question may be kinda stupid, but here goes. Do any surfaces exist for which a parametric form is possible, but cannot be described explicitly due to their highly irregular shape? (Or vice-versa)
  38. D

    Power Series Representation of (1+x)/(1-x)

    Homework Statement For the power series representation of, f(x)=1+x1−x which is 1+2∑from n=1 to inf (x^n), Where does the added 1 in front come from? How do I get to this answer from ∑n=0 to inf (x^n)+∑n=0 to inf (x^(n+1)) Homework Equations The Attempt at a Solution I arrived at ∑n=0 to inf...
  39. T

    Finding the Maclaurin series representation

    Edit: Never mind. Got it. Homework Statement f(x)=\frac { x }{ { (2-x) }^{ 2 } } Homework Equations The Attempt at a Solution I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series. The...
  40. S

    Why is the correction important?

    Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...
  41. L

    Amplitude's time dependence in Heisenberg representation

    A = \langle q_f(t) \mid q_i(t) \rangle = \langle q_{f,H} \mid e^{iH(t_0-t)} e^{-iH(t-t_0)} \mid q_{i,H} \rangle = \langle q_{f,H} \mid q_{i,H} \rangle This means that A is time-independent, and depends only on the reference point ##t_0##. How is it possibly? From Schoedinger picture it...
  42. T

    Operators and the position representation

    I have a question about the formalism of quantum mechanics. For some operator A... \langle x |A|\psi\rangle = A\langle x | \psi \rangle Can this be derived by sticking identity operators in or is it more a definition/postulate. Thanks.
  43. R

    Spinor representation of rotation

    Homework Statement Show the spinor representation corresponding to the rotation through an angle θ about an axis with direction vector n = (n_x, n_y, n_z) has the form: g=exp{-i\frac{θ}{2}(n_x σ_x+n_yσ_y+n_zσ_z)}, σ_{x, y, z} are respectively Pauli matrixHomework Equations h=gxg^{-1}The...
  44. D

    Name of representation of vectors perpendicular to the plane

    Most physicists are familiar with the representation of vectors perpendicular to the plane (\otimes and \odot) which look like the fletching or head of an arrow, commonly used to represent magnetic fields, currents and so on. Can anyone tell me the name of this representation? Does it even...
  45. Fernando Revilla

    MHB De's question at Yahoo Answers (Power series representation)

    Here is the question: Here is a link to the question: Find the first five non-zero terms of power series representation centered at x=0 for the function below.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  46. M

    Linear algebra; find the standard matrix representation

    Homework Statement Find the standard matrix representation for each of the following linear operators: L is the linear operator that reflects each vector x in R2 about the x1 axis and then rotates it 90° in the counterclockwise direction. Homework Equations The Attempt at a Solution So my...
  47. S

    Representation of Lorentz group and spinors (in Peskin page 38)

    I am very confused by the treatment of Peskin on representations of Lorentz group and spinors. I am confronted with this stuff for the first time by the way. For now I just want to start by asking: If, as usual Lorentz transformations rotate and boost frames of reference in Minkowski...
  48. D

    Find parametric representation for the plane

    Homework Statement Plane: 4x−2y+10z =16. Homework Equations The Attempt at a Solution So I've used two parameters, "u" and "v" with x = u and y = v Re-arranging z in terms of "u" and "v": z = 1.6 - 0.4x + 0.2y Hence r(t) = (u , v , 1.6 - 0.4 x + 0.2y) Is this correct?
  49. J

    My problem with the relativity representation on gravity.

    Whenever you see representations of gravity in terms of relativity, you see a planet sitting on a 2d surface of fabric (space) and it is making an indentation, almost as if there another source of gravity pulling it downwards against the fabric. I think this is a poor representation. I mean...
  50. L

    Momentum and wave vector representation

    p=\hbar k So ##dp=\hbar dk## How to define Fourier transform from momentum to coordinate space and from wave vector to coordinate space? I'm confused. Is there one way to do it or more equivalent ways?
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