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. Wrichik Basu

    Java Representation of two-dimensional arrays in the memory

    Arrays declared in java are stored in the memory in two ways: row-major-wise and column-major-wise. As per our teacher, the choice of the storage technique depends on how we enter the array elements. For example, for this code: int arr[][] = new int[10][10]; Scanner kb = new Scanner...
  2. hideelo

    A Extending a linear representation by an anti-linear operator

    When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators...
  3. Luck0

    A Characterizing the adjoint representation

    Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as Ad(U)ta = Λ(U)abtb I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...
  4. Luck0

    A Diagonalization of adjoint representation of a Lie Group

    So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
  5. P

    Dirac delta; fourier representation

    Homework Statement I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ## But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L? Homework...
  6. Milsomonk

    The Dirac equation in Weyl representation

    Homework Statement Compute the antiparticle spinor solutions of the free Dirac equation whilst working in the Weyl representation.Homework Equations Dirac equation $$(\gamma^\mu P_\mu +m)v_{(p)}=0$$ Dirac matrices in the Weyl representation $$ \gamma^\mu= \begin{bmatrix} 0 & \sigma^i \\...
  7. J6204

    Calculating the Fourier integral representation of f(x)

    Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...
  8. Milsomonk

    I Dirac equation solved in Weyl representation

    Hi guys :) I'm just wondering if anyone knows of a book that has the Dirac equation solved in the Weyl basis in it? I'd like to check my method to make sure I'm on the correct lines. Thanks
  9. S

    I Generating a Hilbert space representation of a wavefunction

    Hello, I Have a particle with wavefunction Psi(x) = e^ix and would like to find its Hilbert space representation for a period of 0-2pi. Which steps should I follow? Thanks!
  10. M

    MHB Representing Faults in Manufacturing Departments Using Events

    Hey! :o I am looking the following exercise: A medium-sized company has $n = 3$ manufacturing departments. Faults in the production process can occur in these departments. We have the following events: \begin{align*}&A=\{"\text{All departments work without faults}"\} \\ &B=\{"\text{ no...
  11. RJLiberator

    Valid Representation of Dirac Delta function

    Homework Statement Show that this is a valid representation of the Dirac Delta function, where ε is positive and real: \delta(x) = \frac{1}{\pi}\lim_{ε \rightarrow 0}\frac{ε}{x^2+ε^2} Homework Equations https://en.wikipedia.org/wiki/Dirac_delta_function The Attempt at a Solution I just...
  12. fresh_42

    Insights A Journey to The Manifold SU(2) - Part II - Comments

    Greg Bernhardt submitted a new PF Insights post A Journey to The Manifold SU(2) - Part II Continue reading the Original PF Insights Post.
  13. C

    MHB Series representation for this integral

    I am trying to find a series representation for the following expression $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant. Is there a name for this function? I found a series representation for $$\int_{i=0}^\infty...
  14. T

    I Question about group representation

    After reading some books on Group Theory, I have two questions on group representations (Using matrix representation) with the second related to the first one: 1 - Can we always find a diagonal generator of a group? I mean, suppose we find a set of generators for a group. Is it always possible...
  15. T

    I Spin 1/2 representation of a particle

    A spin 1/2 particle is represented by a spinor while its position is represented by a three-vector. What object should we use to represent such particle if we want to consider both features? That is, what object should we use if we want to consider both spin and space position? It seems there's...
  16. davidge

    I Irreducible representation of SU(2)

    I'm reading a paper on physics where it's said it can be shown that every irreducible representation of ##SU(2)## is equivalent to the one which uses the Ladder Operators. I am a noob when it comes to this subject, but I'd like to know whether or not the proof is easy to carry out.
  17. L

    B Conical Representation of Sphere

    Is Sphere a more generalized form of Cone i.e. formed by 2 dimensional rotation to 360° of a cone? Or is Cone a more generalized form of Sphere since sphere can be formed by rotating about Z axis a zero eccentric planar intersection of a cone? @fresh_42 @FactChecker @WWGD
  18. Runei

    I Complex representation of wave function

    When solving problems, particularly in optics, it is often that we represent the wave-function as a complex number, and then take the real part of it to be the final solution, after we do our analysis. u(\vec{r},t)=Re\{U(\vec{r},t)\}=\frac{1}{2}\left(U+U^*\right) Here U is the complex form of...
  19. L

    B Graphical Representation of a Complex Sphere

    @fresh_42 @FactChecker After thinking, I understood that the answer for this question might make the complex numbers comprehensible for me. My question in detail is as follow Let the equation of a sphere with center at the origin be ##Z1²+Z2²+Z3² = r²## where Z1 = a+ib, Z2 = c+id, Z3 = s+it...
  20. L

    B Dimensional representation of Roots

    If the square root as two coordinate axes in the complex plane, does the cubic root has 3 coordinate axes and so on for nth root? @vanhees71 Can you please explain this?
  21. L

    B Representation of complex of square root of negative i with unitary power.

    Can ##sqrt(-i)## be expressed as a complex number z = x + iy with unitary power?
  22. S

    I Lie Algebra states of a representation

    Hello! I am reading some representation theory/Lie algebra stuff and at a point the author says "the states of the adjoint representation correspond to generators". I am not sure I understand this. I thought that the states of a representation are the vectors in the vector space on which the...
  23. S

    I Normal modes using representation theory

    Hello! I am reading some representation theory (the book is Lie Algebra in Particle Physics, by Georgi, part 1.17) and the author solves a problem of 3 bodies connected by springs forming a triangle, aiming to find the normal modes. He builds a 6 dimensional vector formed of the 3 particles and...
  24. hideelo

    I SO(2n) representation on n complex fields

    If I have a lagrangian which has terms of the form ##\Psi^{\dagger}_\mu \Psi^\mu## then I can decompose the n complex ##\Psi## fields into 2n real fields by ##\Psi_\mu = \eta_{2\mu+1} + i\eta_{2\mu}##. When I look at the lagrangian now it seems to have SO(2n) symmetry from mixing the 2n real...
  25. hideelo

    I Index (killing form ?) in a reducible representation

    In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by Tr(TaR TbR) = T(R)δab I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R1⊕R2. He then states (eqn 70.11) that...
  26. S

    I 6-dimensional representation of Lorentz group

    Hello! I understand that the vector formed of the scalar and vector potential in classical EM behaves like a 4-vector (##A^\nu=\Lambda^\nu_\mu A^\mu##). Does this means that the if we make a vector with the 3 components of B field and 3 of E field, so a 6 components vector V, will it transform...
  27. Y

    MHB Polar Representation of a Complex Number

    Hello all, Given a complex number: \[z=r(cos\theta +isin\theta )\] I wish to find the polar representation of: \[-z,-z\bar{}\] I know that the answer should be: \[rcis(180+\theta )\] and \[rcis(180-\theta )\] but I don't know how to get there. I suspect a trigonometric identity, but I...
  28. B

    A Integral representation of Euler constan

    I am working on the integral representation of the Euler-Mascheroni constant and I can't seem to understand why the first of the two integrals is (1-exp(-u))lnu instead of just exp(-u)lnu. It is integrated over the interval from 1 to 0, as opposed to the second integral exp(-u)lnu which is...
  29. karush

    MHB Find Power Series Representation for $g$: Interval of Convergence

    $\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$ $\textrm{ or Integrating the power series for $f$ perhaps more than once}$ \begin{align*}\displaystyle f(x)&=\frac{1}{1-3x} \\ &=\sum_{k=1}^{\infty} \end{align*} $\textsf{b. Give interval of convergence...
  30. M

    How Does State Space Design Compare to Classical PID Controllers?

    Hello everyone. Iam just Learning about State space representation and controller design and have a fundamental question about the difference between classical Control theory and modern Control theory. I understood the state space is of advantage when dealing with MIMO systems or non-linear...
  31. F

    MATLAB Question about representation of data in Matlab

    Hi everybody; I have plotted a matrix with sea surface temperature's correlation with another variable, the size is (360x180x12); using to plot : figure for i=1:12 subplot(4,3,i);imagescnan(loni,lati,squeeze(double(r4_sat(:,:,i)))'),colorbar; end Now, I want to plot over it another matrix of...
  32. A

    What is parametric representation and how is it used

    Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation. It isn't clearly explained what this means or how you go about...
  33. Vitani11

    Representation of vectors in a new basis using Dirac notation?

    Homework Statement I have a vector V with components v1, v2in some basis and I want to switch to a new (orthonormal) basis a,b whose components in the old basis are given. I want to find the representation of vector V in the new orthonormal basis i.e. find the components va,vb such that |v⟩ =...
  34. S

    A Gauge transformation of gauge fields in the adjoint representation

    In some Yang-Mills theory with gauge group ##G##, the gauge fields ##A_{\mu}^{a}## transform as $$A_{\mu}^{a} \to A_{\mu}^{a} \pm \partial_{\mu}\theta^{a} \pm f^{abc}A_{\mu}^{b}\theta^{c}$$ $$A_{\mu}^{a} \to A_{\mu}^{a} \pm...
  35. F

    I Spinor Representation of Lorentz Transformations: Solving the Puzzle

    I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...
  36. Mr Davis 97

    I Representation of group actions

    I am told that ##\varphi_g (x) = g x g^{-1}## is a group action of G on itself, called conjugacy. However, I am a little confused. I thought that a group action was defined as a binary operation ##\phi : G \times X \rightarrow X##, where ##G## is a group and ##X## is any set. However, this...
  37. M

    A Representation theory of supersymmetry

    I had heard of adinkras but didn't realize that they were meant to play this role. Nor did I realize that the representation theory of supersymmetry is mathematically underdeveloped.
  38. F

    MATLAB Matlab map representation problems

    Hi everybody. I am currently doing a metheorological study of Angola's climate. I can draw the country's shape without much trouble. >> worldmap angola >> load coastlines >> plotm(coastlat,coastlon) But once I have the shape drawn things start to go downhill. I have a vector with the...
  39. Kara386

    Irreducible representation of su(2)

    Homework Statement Using the irreducible representation of ##su(2)##, with ##j=\frac{5}{2}##, calculate ##J_z##, ##exp(itJ_z)## and ##J_x##. Homework EquationsThe Attempt at a Solution There seem to be loads of irreducible representations of ##su(2)## online, but no reference at all to a...
  40. arivero

    I Kronecker product recovering the initial representation?

    In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6. I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?
  41. F

    I Position representation of the state of the system

    Hello Forum, My understanding is that the state of the system is ##|\Psi>##. We can take the inner product between the state ##|\Psi>## and the eigenstates of the position operator ##\hat{x}##: $$<x|\Psi>=\Psi(x)$$ The function ##\Psi(x)## is the wave function we are initially introduced to in...
  42. binbagsss

    Computing representation number quad forms

    Homework Statement ## r_{A} (n) = ## number of solutions of ## { \vec{x} \in Z^{m} ; A[\vec{x}] =n} ## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?) I am solving for the...
  43. F

    I State Representation in QM and Vector Spaces

    Hello Forum, The state of a quantum system is indicated by##\Psi## in Dirac notation. Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis...
  44. P

    B Understanding Fourier Transform for Wavefunction Representation in K Space

    I understand that the Fourier transform to obtain the representation of a wavefunction in k space is $$ \phi(k) =\frac{1}{2\pi}\int{dx \psi(x)e^{-ikx} } $$ and that $$p=\bar{h} k$$ But why then is $$\phi(p) =\frac{\phi(k)}{\sqrt{\bar{h}}} $$ Many thanks in advance :)
  45. K

    Not understanding this series representation

    [mentor note: thread moved from non-hw forum to here hence no homework template] Can someone explain to me how it is that $$\sum_{n=a}^b (2n+1)=(b+1)^2-a^2$$ I thought it would be $$\sum_{n=a}^b (2n+1)=(2a+1)+(2b+1)$$ but I am clearly very wrong. I would greatly appreciate any help.
  46. binbagsss

    A Representation number via quad forms of theta quadratic form

    ##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ## ##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##, where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]## where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive...
  47. M

    Hydrogen in Magnetic Field, Interaction Representation

    The hydrogen is placed in the external magnetic field: $$ \textbf{B}=\hat{i}B_1 cos(\omega t) + \hat{j} B_2 sin(\omega t) + \hat{k} B_z ,$$ Using the relation ## H = - \frac{e\hbar}{2mc} \mathbf \sigma \cdot \mathbf B ##, then I got the form $$ H = H_0 + H' , $$ where $$ H'= - \frac{e...
  48. karush

    MHB 206.r2.11find the power series representation

    $\tiny{206.r2.11}$ $\textsf{find the power series represntation for $\displaystyle f(x)=\frac{x^7}{3+5x^2}$ (state the interval of convergence), then find the derivative of the series}$ \begin{align} f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...
  49. W

    I Spectral representation of an incompressible flow

    Hi PH. Let ##u_i(\mathbf{x},t)## be the velocity field in a periodic box of linear size ##2\pi##. The spectral representation of ##u_i(\mathbf{x},t)## is then $$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary...
  50. StudentOfScience

    Kleppner - Example 1.18: Representation of Position Vector

    Homework Statement Bead on spoke: constant speed ##u## along spoke it starts at center at ##t=0## angular position is given by ##\theta=\omega t##, where ##\omega## is a constant Homework Equations ## \frac{d\hat r}{dt} = \dot \theta \hat \theta ## (1) ## \frac{d\hat \theta}{dt} = -\dot...
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