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. T

    A Question about the irreducible representation of a rank 2 tensor under SO(3)

    When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...
  2. pellis

    I Calculating group representation matrices from basis vector/function

    Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...
  3. JD_PM

    Matrix representation of a linear mapping

    I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...
  4. PrimoCosta

    Eletromagnétic field representation

    Hello so my doubt is when i are teaching about eletromagnétic filds the frist time a do a representatios is of a full field i Draw the two waves, ume perpendicular to another, and istill i know that's not the best representatios because i should have to rolate the wave while a draw. Later in...
  5. S

    Sakurai 3.21, Cartesian eigenbasis representation

  6. H

    I Matrix Representation of the Angular Momentum Raising Operator

    In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...
  7. E

    Quick question about the representation of error on a graph

    Hello I hope you are all very well. I did a graph for a physics practical work and I would like to put the "error bar" (I don't know if it's called like this, I'm sorry). Something like this: I just don't know how long the margin should be. If someone could clarify to me the way to do the...
  8. Jason Bennett

    Lorentz algebra elements in an operator representation

    1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
  9. e101101

    Complex representation of a wave

    Homework Statement: Hi there, I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an...
  10. T

    I Anticommutators of the spin-1 representation

    The Pauli matrices of the spin-1 representation are given by: ##T_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}##, ##T_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}## and ##T_{3}=\begin{pmatrix} 1 & 0 & 0 \\ 0...
  11. filip97

    A (A,A) representation of Lorentz group-why is it tensor?

    Why representation of Lorentz group of shape (A,A) corespond to totally symmetric traceless tensor of rank 2A? For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ? See Weinberg QFT Book Vol.1 page 231.
  12. DarMM

    A Representation of Super-Quantum probability

    Classical probability theory can be represented with measure spaces and functions over them. Quantum Probability is given as the theory of Hilbert spaces and operators over them. Both more abstractly are handled by the theory of C*-algebras and their duals. However I know of no structure for...
  13. cianfa72

    Linear 2-port network representation

    Hi, in the context of linear two-ports networks my (italian) textbook says that if its internal structure consists of passive one-port components with no independent current/voltage generators then the following (implicit) linear equations based representation always exist: ##\left[A\right] V...
  14. J

    I Matrix Representation of an Operator (from Sakurai)

    Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck. I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down...
  15. velikh

    B On the representation of integers?

    Let (a, b, c) be some arbitrary positive integers such that: (q2^0 + q2^1+ . . . + q2^x), (q2^0 + q2^1+ . . . + q2^y), (q2^0 + q2^1 + . . . + q2^z), where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n). In the case if and only if q = 2, we accept the following notation : [(q-1)2^0...
  16. PainterGuy

    I What does it mean when an integral is evaluated over a single limit?

    Hi, A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component. For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series...
  17. N

    A Block Diagonalization - Representation Theory

    How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
  18. R

    Prove q-ary representation of n*q

    ##nq = q(d_1 + d_2 q + d_3 q^2 + \dots + d_k q^{k-1})## ##= d_1 q + d_2 q^2 + d_3 q^3 + \dots + d_k q^k## ##= d_k d_{k-1} \dots d_1 0## Can someone please explain how to get from line two to line three. This is instructors solution and not sure I understand. Thanks!
  19. A

    I Invariant symbol implies existence of singlet representation

    I don't understand what the last paragraph of the attached page means. Why does the Kronecker delta being an invariant symbol mean that the product of a representation R and its complex conjugate representation has the singlet representation with all matrices being zero? Doesn't the number...
  20. TheBigDig

    Spin Annhilation and Creator Operators Matrix Representation

    Homework Statement Given the expression s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> obtain the matrix representations of s+/- for spin 1/2 in the usual basis of eigenstates of sz Homework Equations s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> S_{+} = \hbar...
  21. YoungPhysicist

    Physical representation of bead sort

    Bead sort is a not practical sorting algorithm that suppose to have a time complexity of ##O(\sqrt{n})##in the real world. I thought it would be great if a device that can take numbers from the computer,then turn it into analog stuff that the device can read,drop the beads,and return it...
  22. J

    A Fields transforming in the adjoint representation?

    Hi! I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or...
  23. J

    Proving Dn with Involutions: Group Representation Homework

    Homework Statement let n ≥ 2 Show that Dn = < a,b | a2, b2, (ab)n> Homework EquationsThe Attempt at a Solution I see that a and b are involutions and therefore are two different reflections of Dn. If we set set b = ar where r is a rotation of 2π/n And Dn = <a,r | a2, rn, (ab)2 > I am unsure...
  24. Hans de Vries

    A New Covariant QED representation of the E.M. field

    90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field. 1 - Definition of the...
  25. mnb96

    A Representation of elements of the Grassmannian space

    Hi, I am studying some material related to Grassmannians and in particular how to represent k-subspaces of ℝn as "points" in another space. I think understood the general idea behind the Plücker embedding, however, I recently came across another type of embedding (the "Projection embedding")...
  26. A

    I Adjoint Representation Confusion

    I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this: My understanding of this question is that, given a...
  27. K

    Vector representation of Lorentz group

    Homework Statement In this problem, we'll construct the ##(\frac{1}{2},\frac{1}{2})## representation which acts on "bi-spinors" ##V_{\alpha\dot{\alpha}}## with ##\alpha=1,2## and ##\dot{\alpha}=1,2##. It is convential, and convenient, to define these bi-spinors so that the first index...
  28. K

    I Ladder operators and SU(2) representation

    Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...
  29. CharlieCW

    General Irreducible Representation of Lorentz Group

    This one may seem a bit long but essentially the problem reduces to some matrix calculations. You may skip the background if you're familiar with Lorentz representations. 1. Homework Statement A Lorentz transformation can be represented by the matrix...
  30. H

    B Is there a way to find the integer representation of a real number?

    Is there a way to find the integer of a real number? Of course without using the [x] function. What I am looking for here is an algebraic formula.
  31. S

    State Space representation of an inverted Pendulum

    Homework Statement I am trying to learn how to create state space representations. I am using this link to study. http://www2.ensc.sfu.ca/people/faculty/saif/ctm/examples/pend/invpen.html Could someone explain how to get the matrix A from my equations? Homework EquationsThe Attempt at a...
  32. Ramtin123

    A Three dimensional representation of ##U(1)\times SU(2)##

    Consider a three dimensional representation of ##U(1)\times SU(2)## with zero hypercharge ##Y=0##: $$ L= \begin{pmatrix} L^+ \\ L^0 \\ L^- \end{pmatrix} $$ Then the mass term is given by [1]: $$ \mathcal{L} \supset -\frac m 2 \left( 2 L^+ L^- +L^0 L^0 \right) $$ I am wondering where the...
  33. Spin One

    I Solving the Representation of d/dq: Mistake in Reasoning?

    When considering the position representation of a system with one degree of freedom endowed with canonical co-ordinates and momenta q and p, Dirac deduces that: which is the representation of d/dq in "matrix" form. But the derivative of a delta function is, I assume (from the definition of the...
  34. C

    I Lorentz Group: Tensor Representation Explained

    I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...
  35. T

    MHB What is the minimal dimension of a complex realising a group representation?

    This question is inspired by one question, which was about representations that can be realized homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral representations of groups arise via homology from a group acting on a simplicial complex that is...
  36. G

    I Two-dimensional vector representation

    Hi, Is there a method to represent a known two-dimensional vector field w of two coordinates x and y with zero divergence and non-zero curl as $$ \vec{w}(x,y) = a \nabla b \, , \hspace{4mm} \nabla \cdot \vec{w} = 0 \, , \hspace{4mm} \nabla \times \vec{w} = f(x,y) \, ?$$ How would one proceed to...
  37. MathematicalPhysicist

    A Why is gluon's representation 8 in ##SU(3)_c## and not 1?

    I understand that because the rep. of quark is 3 and antiquark is ##\bar{3}## and we have ##3\otimes \bar{3} = 8\oplus 1##, that a gluon should be 8 or 1, but which one? and why? Thanks!
  38. hideelo

    A Does an irreducible representation acting on operators imply....

    Ok, so my question is "Does an irreducible representation acting on operators imply that the states also transform in an irreducible representation?" and what I mean by that is the following. If I have an operator transforming in an irreducible transformation of some group, I get a corresponding...
  39. frostysh

    What representation of time did Poincare use in his work in 1905?

    Recently have a discussion with a Scientist (myself is not a Scientist, but trying to become :P), about priority of Einstein and Poincarè for Special Relativity Theory invention in Science. Those, what actually contribution of Einstain, what actually contribution of Poincarè and what...
  40. Mutatis

    Write the matrix representation of the raising operators....

    Homework Statement Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##. Homework Equations I know how to use all the above operators projecting them on...
  41. Luck0

    A Diagonalizing Hermitian matrices with adjoint representation

    Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...
  42. stevendaryl

    A Coordinate representation of a diffeomorphism

    I'm trying to understand diffeomorphisms, and I thought I basically understood them, but when I tried to work out a problem I created for myself, I realized I didn't know how to answer it. So let's consider a diffeomorphism generated by a vector field ##V##. If ##X## is a point on our manifold...
  43. zexxa

    Canonical ensemble of a simplified DNA representation

    Question Form the canoncial partition using the following conditions: 2 N-particles long strands can join each other at the i-th particle to form a double helix chain. Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open" An "open" link gives the...
  44. diegzumillo

    I Breaking down SU(N) representation into smaller groups

    Hi all I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of...
  45. R

    Matrix representation relative to bases

    Homework Statement Please see attached file. I'm not quite sure if I'm on the right track here. I think the basis for F is throwing me off as well as T(f). Please advise. Thanks! Homework EquationsThe Attempt at a Solution
  46. N

    SU(3) Cartan Generators in Adjoint Representation

    I am trying to work out the weights of the adjoint representation of SU(3) by calculating the 2 Cartan generators as follows: I obtain the structure constants from λa and λ8 using: [λa,λb] = ifabcλc I get: f312 = 1 f321 = -1 f345 = 1/2 f354 = -1/2 f367 = -1/2 f376 = 1/2 f845 = √3/2 f854 =...
  47. S

    I Representation Theory clarification

    Hello! I am reading some things about representation theory for SU(n) and I want to make sure I understand it properly. I will put an example here and explain what I understand out of it and I would really appreciate if someone can tell me if it is right or not. So for SU(2) we have ##2 \otimes...
  48. N

    Adjoint representation of SU(2)

    Homework Statement [/B] I am looking at this document. http://www.math.columbia.edu/~woit/notes3.pdf Homework Equations [/B] ad(x)y = [x,y] Ad(X) = gXg-1The Attempt at a Solution [/B] I understand how ad(S1) and X is found but I don't understand what g and g-1 to use to find Ad(X). Also...
  49. M

    Representation of Rotating and Translating Body

    Hello Everyone.. Consider a small moving mass following a spiral path away from center and the body is rotating about its center as well. At some time the body impacts with a relatively bigger mass and transfers some of its energy to it. If we want to make an equivalent system with only linear...
  50. jk22

    B Gravity and spin 2 representation

    I'm not at all involved in QG but from far away I noticed : Spin 2 representations are 5x5 matrices. But in gravity what mathematical objects are quantized ? If it's the metric then it's a 4x4 matrix so that cannot be that. Or : how does quantization reveal a 5x5 matrix ?
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