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. Timakki

    Mathematical representation of probability

    I am having some difficulties in understand the convention of probability like P(A|E) ... And I am not able convert the questions into this for .I can solve those questions but can someone help me to understand this topic .And also multiplication theorem .
  2. karush

    MHB S4.12.9.13 find a power series representation

    $\tiny{s4.12.9.13}$ $\textsf{find a power series reprsentation and determine the radius of covergence.}$ $$\displaystyle f_{13}(x) =\frac{1}{(1+x)^2}=\frac{1}{1+2x+x^2}$$ $\textsf{using equation 1 }$ $$\frac{1}{1-x} =1+x+x^2+x^3+ \cdots =\sum_{n=0}^{\infty}x^n \, \, \left| x \right|<1$$...
  3. R

    Is pixel a representation of intensity for light at a given location?

    Is pixel a representation of intensity for light at a given location?
  4. binbagsss

    I Index Notation, Covector Transform Matrix Rep

    Just a couple of quick questions on index notation, may be because of the way I'm thinking as matrix representation: 1) ##V^{u}B_{kl}=B_{kl}V^{u}## , i.e. you are free to switch the order of objects, I had no idea you could do this, and don't really understand for two reasons...
  5. lfdahl

    MHB Prove, that the decimal representation: pk = 11111....1 exists for every prime, p > 5.

    Let $p$ be a prime number exceeding $5$. Prove that there exists a natural number $k$ such that each digit in the decimal representation of $pk$ is $1$ : $pk = 1111...1$
  6. whatisgoingon

    Matrix representation of a quantum system

    Homework Statement I have to find the matrix system of Sx, Sy , and Sz using the given information: 190899[/ATTACH]'] Homework EquationsThe Attempt at a Solution for attempting Sx: Ignoring the ket at the bottom, I would get Sx|+> = +ħ/2[[0,1],[1,0]] but my question here is, does the ket at...
  7. J

    B Why can't one metre be more than one metre?

    Hello, Recently, I have been trying to work on some philosophy that I am developing, and the subject of measurement has come up. My question goes a bit like this: Suppose that it turned out that when you measure point A to point B as exactly 100cm (one metre), there are actually three...
  8. Marcin H

    Binary Addition - 2's Compliment Representation

    Homework Statement Homework Equations Binary Arithmetic The Attempt at a Solution Is this the right way to do these problems? If I see a binary 1 at the beginning of the binary number do I just do the 2's compliment of that and add them? I am not too sure what it means by assume numbers...
  9. Twigg

    A Is representation theory worthwhile for quantum?

    Recently I read some comment on Sakurai's book (which I have not read) that the writer of said comment didn't understand part of the text until they understood irreducible representations. I do not know to what they were referring, but it piqued my interest in representation theory. My question...
  10. RJLiberator

    Coordinate representation of vectors?

    Homework Statement Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book. (1.16)A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right] Note: Here, I use * to...
  11. A

    Matrix representation of certain Operator

    Homework Statement Vectors I1> and I2> create the orthonormal basis. Operator O is: O=a(I1><1I-I2><2I+iI1><2I-iI2><1I), where a is a real number. Find the matrix representation of the operator in the basis I1>,I2>. Find eigenvalues and eigenvectors of the operator. Check if the eigenvectors are...
  12. Tareq Naushad

    Medical How does our brain create a summarized/average representation

    When I try to imagine face of some persons like relative,friend, celebrities etc. I can visualize them in my mind. I wonder how do brain decide what to show to me in my mind about that person? I might have seen the person at his/her various ages, various dresses, various places. While...
  13. S

    I State of a Generator in Representation Theory

    Hello! I am reading something about representation theory (just started) and I encountered this: "We will denote the state in the adjoint representation corresponding to an arbitrary generator ## X_a ## as |## X_a ##>". What is the state of a generator in a certain representation? Thank you!
  14. S

    I No problem, it's always good to have multiple sources!

    Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...
  15. N

    I Bases for SU(3) Adjoint representation

    What are the bases for the adjoint representation for SU(3)?
  16. T

    Taylor series representation for $$ \frac{x}{(1+4x)^2}$$

    Homework Statement Find a power series that represents $$ \frac{x}{(1+4x)^2}$$ Homework Equations $$ \sum c_n (x-a)^n $$ The Attempt at a Solution $$ \frac{x}{(1+4x)^2} = x* \frac{1}{(1+4x)^2} $$ since \frac{1}{1+4x}=\frac{d}{dx}\frac{1}{(1+4x)^2} $$ x*\frac{d}{dx}\frac{1}{(1+4x)^2}...
  17. I

    I Finding components in momentum representation

    Dear all, I am trying to understand how they get they get the following components: $$c_\textbf{p} = \langle \textbf{p} | \psi\rangle = \int \frac{d\textbf{r}}{\sqrt{V}}e^{-\frac{i}{\hbar}\textbf{p}\cdot\textbf{r}}\psi(\textbf{r})$$ Where ##|\textbf{p}\rangle## are the plane waves $$...
  18. N

    I Adjoint representation of SU(3)

    Not sure if this is the correct forum but here goes. I am trying to prove [Ta,Tb] = ifabcTc Where (Ta)bc = -ifabc and fabcare the structure constants for SU(3). I picked f123 and generated the three 8 x 8 matrices .. T1, T2 and T3. The matrices components are all 0 except for, (T1)23 = -i...
  19. A

    I Parametric Representation of a Solution Set

    How did he found x = 1, y = 0, z = 0 and x = 1, y = 1, z = 2?
  20. H

    Finding a matrix representation of a Hamiltonian.

    Homework Statement The Hamiltonian H for a certain physical quantum mechanical system has three eigenvectors {|v1>, |v2>, |v3>} satisfying: H|vj> = (2-j)a|vj> Write down the matrix representing H in the representation {|v1>, |v2>, |v3>} . Homework EquationsThe Attempt at a Solution I though...
  21. P

    Linear transformation representation with a matrix

    Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...
  22. Augbrah

    How Do Representations of SO(5) Decompose into SO(4)?

    Homework Statement Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 Homework EquationsThe Attempt at a Solution [/B] I understand that 5 is rep of SO(5) corresponding to Dynkin labels (1, 0). 1 is...
  23. S

    Matrix representation of function composition

    Am I on the right path here? 1. Homework Statement i. Prove that ##T_{a}## and ##T_{b}## are linear transformations. ii. Compose the two linear transformations and show the matrix that represents that composition. 2. The attempt at a solution i. Prove that ##T_{a}## and ##T_{b}## are linear...
  24. KT KIM

    I Matrix Representation of Linear Transformation

    This is where I am stuck. I studied ordered basis and coordinates vector previous to this. of course I studied vector space, basis, linear... etc too, However I can't understand just this part. (maybe this whole part) Especially this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of...
  25. nfcfox

    Why Is the Power Series Automatically Centered at x=2?

    Homework Statement http://imgur.com/12LbqWL Part b Homework EquationsThe Attempt at a Solution Since it says the first four terms, not nonzero, the first four terms would be 0-(1/3-0)+2/9(x-2)-1/9(x-2)^2 I'm confused when it says I need to find these for x=2... Do I just plug in x=2 now and...
  26. D

    Derive the representation of the momentum acting on a field

    Homework Statement consider the space-time transformation of translation xμ → x'μ = xμ + aμ where xμ is a point in space-time and aμis a constant 4-vector. Assuming translations are generated by the operator U=e-iPμaμ acting on fields Φ(x), derive the representation of Pμ on the field Φ(x)...
  27. D

    I Momentum operator on positon/momentum representation

    Hi. I have come across the following step in a derivation of the harmonic oscillator groundstate wavefunction using ladder operators ∫ <x | p | p><p | o > dp = ∫ p<x | p><p | o > dp = -iħ d/dx ∫ <x | p><p | 0>dp I am confused about how the -iħ d/dx arises. I thought the p produced when the p...
  28. C

    I Adjoint representation and the generators

    Given that ##g T_a g^{-1} = D^b_a T_b## one can show that the generators in the adjoint representation of a group ##G## are the structure constants of the lie algebra satisfied by the ##T_a##. Write ##g## infinitesimal, so that ##g = 1 + \mathrm {i} \alpha^a T_a## and ##D^c_a = \delta^c_a + i...
  29. C

    Bilinears in adjoint representation

    Homework Statement [/B] 1)Show that the kinetic term for a Dirac spinor is invariant under the symmetry group ##U(N) \otimes U(N)## 2) Show that if ##T_a## are the generators of ##O(N)##, the bilinears ##\phi^T T^a \phi## transform according to the adjoint representation. Homework Equations...
  30. I

    Courses Representation theory or algebraic topology

    Hello everyone, I'm a undergraduate at UC Berkeley. I'm doing theoretical physics but technically I'm a math major. I really want to study quantum gravity in the future. Now I have a problem of choosing courses. For next semester, I have only one spot available for either representation theory...
  31. NihalRi

    Use geometric series to write power series representation

    Homework Statement Complete the proof that ln (1+x) equals its Maclaurin series for -1< x ≤ 1 in the following steps. Use the geometric series to write down the powe series representation for 1/ (1+x) , |x| < 1 This is the part (b) of the question where in part (a)I proved that ln (1+x)...
  32. B

    MHB Parametric representation of a line

    I am give the following curve r(t) = (t+1,0.5(1-t),0) where t ranges from -1 to 1. I am now trying to derive a new parametric representation of this line segment using the arc length as the parametric variable. I have integrated r'(t) from -1 to 1 and found that the length of the segment ranges...
  33. I

    Position representation of coherent state and time evolution

    Homework Statement I ended up solving the problem as I was typing it up, I am posting what I did anyway as it took so long to type and might be useful to someone else. I am trying to figure out the position representation of a coherent state and it's time evolution. I should be getting a...
  34. evinda

    MHB How Do Cylindrical and Spherical Coordinates Represent the Same Point?

    Hello! Suppose that a point has cylindrical coordinates $(r, \theta, z)$ where $r$ is not zero. Describe all other cylindrical coordinates of that point. Suppose $(R, \theta_1, \phi)$ and $(R, \theta_2, \phi)$ are two representations of the same point in spherical coordinates. Is it true that...
  35. KastorPhys

    I Representation between State Vector & Wave Function

    By the Principle of Superposition, a state vector can be defined as pic.01 also, the state vector can represent a wave function in a continuous case as pic.02 My (1) question is, in pic.03, why the state vector can be pulled out from the integral? I have an idea but I think it should be...
  36. C

    How do I evaluate <x> with the k-space representation?

    Homework Statement Given the following k-space representation of the wave function: Ψ(k,t) = Ψ(k)e-iħk2t/2m use the wave number representation to show the following: <x>t=<x>0 + <p>0t/m <p>t=<p>0 Homework Equations <x>=∫Ψ*(x,t)xΨ(x,t)dx <p>=∫Ψ*(x,t)(-iħ ∂/∂x)Ψ(x,t)dx The Attempt at a...
  37. omidaut

    Representation of Dirac Spinors

    Hi there, I have a question with its answer, but, still I don't understand it. Can anybody help me in explaining it? Thanks.
  38. G

    Adjoint representation of Lorentz group

    Hey, There are some posts about the reps of SO, but I'm confused about some physical understanding of this. We define types of fields depending on how they transform under a Lorentz transformation, i.e. which representation of SO(3,1) they carry. The scalar carries the trivial rep, and lives...
  39. H

    Operator r is a diagonal matrix in position representation

    What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that? Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)
  40. Raptor112

    Matrix Representation for Combined Ladder Operators

    Due to the definition of spin-up (in my project ), \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 2 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} as opposed to \begin{eqnarray} \sigma_+ = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix} \end{eqnarray} and the annihilation operator is...
  41. Rumo

    Weyls representation of a propagating (z-v*t) spherical wave

    Hello! The following wave solves the 3D wave equation: $$ \frac{\sin\left(k\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}\right)}{\sqrt{x^2+y^2+\frac{(z-vt)^2}{1-\frac{v^2}{c^2}}}}\cos\left(w\frac{t-\frac{vz}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\right) $$ This is a propagating standing spherical...
  42. N

    Representation of (x-7) and (x+1)

    Mod note: Changed the title and this post to reflect the situtation in the diagram the representation of the inequality (x-7)(x+1) ≤ 0 makes no sense to me . can someone explain this . the diagram is attached
  43. Hijaz Aslam

    Euler Representation of complex numbers

    I am bit confused with the Eueler representation of Complex Numbers. For instance, we say that e^{i\pi}=cos(\pi)+isin(\pi)=-1+i0=-1. The derivation of e^{i\theta}=cos(\theta)+isin(\theta) is carried out using the Taylor series. I quite understand how ##e^{i\pi}## turns out to be ##-1## using...
  44. Harry Mason

    Understanding the phase-space representation of Ensembles

    Hello , i have some troubles with basics concepts of statistical mechanics. I feel confortable with the general idea of an Ensamble, a collection of copies of the physical system which differs from each other due to microscopic differences and having the same macroscopic behavior. I'm ok also...
  45. H

    Matrix representation of operators

    Let the operators ##\hat{A}## and ##\hat{B}## be ##-i\hbar\frac{\partial}{\partial x}## and ##x## respectively. Representing these linear operators by matrices, and a wave function ##\Psi(x)## by a column vector u, by the associativity of matrix multiplication, we have...
  46. H

    Matrix representation of an operator with a change of basis

    Why isn't the second line in (5.185) ##\sum_k\sum_l<\phi_m\,|\,A\,|\,\psi_k><\psi_k\,|\,\psi_l><\psi_l\,|\,\phi_n>##? My steps are as follows: ##<\phi_m\,|\,A\,|\,\phi_n>## ##=\int\phi_m^*(r)\,A\,\phi_n(r)\,dr## ##=\int\phi_m^*(r)\,A\,\int\delta(r-r')\phi_n(r')\,dr'dr## By the closure...
  47. G

    Lie Group v Lie algebra representation

    Hi y'all, This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer. I have a Lie group homomorphism \rho : G \rightarrow GL(n...
  48. D

    Understanding Karnaugh Maps: Two Ways of Representation

    Well, I've recently been studying karnaugh maps and I've noticed there's two sorts of ways to represent them. In my learning materials, sometimes they are expressed in one way,sometimes in another way. For example: http://puu.sh/luJBp/4b2c878122.png Now, what I don't understand is with this...
  49. Vinay080

    Motivation for geometrical representation of Complex numbers

    I am seeing in "slow motion" the development of vectorial system. I am reading the book "A History of Vector Analysis" (by Michael J.Crowe); it seems from my acquaintance that the vector concept came from the quaternions concept; and the quaternions concept came from the act of search for...
  50. Msilva

    Finding a matrix representation for operator A

    I need to find a matrix representation for operator A=x\frac{d}{dx} using Legendre polinomials as base. I would use a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx, but I have the problem that Legendre polinomials aren't orthonormal \langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}. I...
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