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

    Modeling mathematical representation for music signal

    Hello there I am sorry if this sounds silly or out dated i want to ask anyone, how can i model the mathematical representation from music signal(time domain). lets say i recorded the signal from guitar through my music production software and print screen that signal as shown here...
  2. K

    Use Differentiation to Find a Power Series Representation for:

    Homework Statement for a.) f(x) =1/ ( (1+x)^2 ) what is the radius of convergence? b.) Use part a.) to find a power series for f(x)=1/ ( (1+x)^3) c.) Use part b.) to find a power series for f(x) =x^2 /( (1+x)^3) Homework Equations I want to check my work. I used properties of functions...
  3. S

    Position representation of a wavefunction - technical issue

    Hey, I have been churning the idea of Dirac notation around in my head and I am thinking about the position and momentum basis representation of a wavefunction in a Hilbert space. Wikipedia mentions the following in the article 'Bra-ket notation' under the heading 'Position-space wave...
  4. M

    Matrix Representation Question

    Homework Statement This is a linear algebra matrix representation problem I have been trying to solve. I seem to keep getting it wrong, so I was hoping I could get some help. The linear operator T: P2-->P2 is defined by T(P(x)) = xP'(x)-P(x). B={1,x,x2}, and B'={x,1+x,-1+x2} are two...
  5. J

    RThe canonical representation phi (measure theory) (Royden)

    RThe "canonical representation phi" (measure theory) (Royden) Homework Statement I need some help understanding the canonical representation of phi as described on p. 77 of Rodyen's 3rd edition. I've transcribed it below for those of you who don't own the book. Homework Equations The...
  6. L

    How does spin work in the position representation?

    When you have an ordinary (read: spin zero) particle with a quantum state |ψ>, the corresponding position-space wave function <x|ψ> is an ordinary scalar function of position. How do things work when the particle has spin? In that case its quantum state will live in the tensor product of two...
  7. H

    Symmetric vector to tensor representation?

    Hi all! I have a discrete 2D vector field with a particular characteristic: At every point, instead of having a single vector, I have two vectors which are in the opposite direction. For example, at point p(x,y)=p(0,0) I have two vectors: v1(1,1) and v2(-1,-1). And so on for all points. I...
  8. T

    Integral representation of the delta function

    Homework Statement http://gyazo.com/7b2a903b6b3165595b8766d3540f43d9 What is this really saying? I can see that a functino is the inverse Fourier transform of the Fourier transform... and it doesn't matter which way round you integrate. Is that all it's saying. What's the difference...
  9. J

    How to avoid rounding off in binary representation

    Homework Statement Is the fundtion defined f:P(\mathbb{N}) \to [0,1] by f(X) = 0.a_{1} a_{2} \dots in binary representation where a_{k}=1 if k\in X and otherwise 0 one-to-one? (*note: N does not have 0) If not, can you change bit so that the changed funtion becomes one-to-one...
  10. Q

    Finding the 3x3 Matrix Representation of SU(2)

    Hi all, do you know where i can find the 3x3 matrix representation of SU(2)? Which means basically rotation matrices for particles of spin 1. Thanks!
  11. fluidistic

    Thermodynamics, entropy representation problem

    Homework Statement The problem is taken from Callen's book (page 36). Find the three equations of state in the entropy representation for a system with the fundamental equation u =A\frac{s^{5/2}}{v^{1/2}}. Show by a diagram (drawn to abitrary scale) the dependence of temperature on volume...
  12. Q

    Momentum space representation for finite lattices - continued

    I have been banned, maybe my nickname was not so kind. I let the topic continue here. I report my last comment: "Ok, I got the point. thanks for replying! It's just a change of basis that under boundary condition diagonalize the Hamiltonian. But then a subtle point: In order for...
  13. B

    Momentum space representation for finite lattices

    Hi all, I have a question. For sure the momentum representation used in solid state physics works for infinite lattices or periodic ones. But when it comes to finite lattice, i.e. 100 sites, can the momentum representation be used? What are the errors? Where does this fail? Thanks for...
  14. Math Amateur

    Representation Theory of Finite Groups - CH 18 Dummit and Foote

    I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...
  15. H

    Coordinate System of Coupled Oscillators and 4D Phase Space representation

    Coordinate System of Coupled Oscillators and "4D" Phase Space representation So, I've modeled the interaction between two cantilever beams with the kinetic and potential energies shown in the above figure. The cantilevers are very stiff and have a small oscillation amplitude, so they can be...
  16. M

    Representation theory and totally symmetric ground state?

    Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...
  17. A

    Working w/ Complex representation E-field

    Often a time varying E-field is represented in complex format. I have a simple E-field (uniform in space) given by \vec{E}(t)=E_o\cos(\omega t)\cdot\hat{k} or equivalently, the real part of \vec{E}(t)=E_o e^{\omega t}\cdot\hat{k}. We know the potential is the negative gradient of the...
  18. L

    Helps on understanding different representation transformations

    Hi,all, I m an undergrades and I am suffering on understanding the different representation transformations, namely from schrodinger picture to interaction picture tupically, my lecturer didn't state which representation he was using and I m so confused, any helps would be great. Shall I bring...
  19. A

    Complex representation of electric field

    Confused.. please help! Often when an electric field varies sinusoidally with time, it is represented as a complex number. Say, \vec{E}(t)=A\cos(t) \cdot \hat{k} We know at any time, the magnitude of E is A\cos(t). Alternatively the same vector E is understood to be the real part of the...
  20. Z

    Basic Symmetric Group Representation Question

    If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the...
  21. T

    What does the frequency representation of a function show?

    I don't understand what electrical engineers mean by the frequency of a signal... Frequency is the inverse of the period, but they speak of the frequency of non-periodic signals. I know that I can derive the function using the Fourier transform, I just don't understand what frequency means in...
  22. L

    Why not diffeomorphism group representation theory?

    For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei...
  23. Telemachus

    Silly doubt: ideal gas eq. of state entropic representation

    Hi there. I have a silly doubt about the entropy of mixing for ideal gases The entropy of mixing is this (Eq. [1]): S_{mix}=\sum_j N_j s_{j0}+\left (\sum_j N_j c_j \right) R \ln{\frac{T}{T_0}}+\sum_j N_j R \ln {\frac{V}{N_jv_0}} Now I don't know what identity the book uses to rewrite this on...
  24. Telemachus

    Silly doubt about thermodynamics: molar representation

    Well, I have a doubt about something that I've found in the book. It's really silly, but it's been bothering me for a while, so perhaps you can help me to understand this. As you should know, the fundamental equation of a system can be represented in the entropic representation as a function...
  25. F

    Representation Theory and Particle Theory

    I am familiar with the representation theory of finite groups and Lie groups/algebra from the mathematical perspective, and I am wondering how quantum mechanics/quantum field theory uses concepts from representation theory. I have seen the theory of angular momentum in quantum mechanics, and I...
  26. E

    Lorentz Boosts in Group Representation (from Weinberg)

    Alright, so excuse my ignorance, but I have no idea why the choice he uses for boosts is "convenient" Just to make sure everyone is on the same metric etc etc. Weinberg uses (-,+,+,+) with gamma defined traditionally and God-given units He requires that transformations..(oh my,,,how...
  27. T

    Nonsensical representation of gravity

    Greetings all, I am the newest addition of the PF community and let me express my excitement of finally tracking down this intelectual family and hopefully making myself eventually an equal member of it. Now, I will not waste too much of your time let us get down to this PFbuisness=pleasure...
  28. S

    Two's Complement Representation

    Homework Statement When determining the two's complement of a decimal number, one converts the decimal to binary, finds the complement of each individual digit, adds one, and there we have it. For example, -4=1100--->0011+1=0100=+4 However, when I apply the same logic for -99, I don't...
  29. N

    What is adjoint representation in Lie group?

    Please teach me this: What is the adjoint representation in Lie group? Where is the vector space that the ''elements of the group'' act on in this representation(adjoint representation)? Thank you very much for your kind helping.
  30. R

    Finite Groups and Three Dimensional Pseudo Real Representations: An Exploration

    Can you please give an example for a finite group with a three dimensional pseudo real representation? I can find examples of finite groups with 2, 6 and 8 dimensional pseudo real representations, but couldn't find any with a three dimensional pseudo real rep. Is there some theorem that states...
  31. N

    Circuit representation of cell

    Hello! I am confused about this circuit representation. I am not familiar with how it works. I was under the impression charge cannot flow across a capacitor (a wire is required). So how does charge move in the following circuit? Not across the battery - since the charge is supposed to be...
  32. T

    Spinors and Lorentzgroups: representation of the complete Lorentzgroup?

    Homework Statement Hi, This question is about Lorentzgroups. In my course of Relativity, we've seen a very little about representations of complete Lorentz groups but there are two little exercises, which we can do, but I do not understand what should be checked, not even how to start this...
  33. T

    Linear Algebra (Matrix representation of linear operators)

    Homework Statement Determine [T]β for linear operator T and basis β T:((x1; x2]) = [2x1 + x2; x1 - x2] β = {[2; 1], [1; 0]} Homework Equations Now that would be MY question :rolleyes: The Attempt at a Solution Well the answer is [1, 1; 3, 0], but i have no idea what I'm even...
  34. P

    Graph representation of particles

    Howdy. I'm interested in finding connections between graph theory and particle physics, particularly where particles are represented as nodes on a graph. It's my understanding that this is not a unique idea. Would anyone know where I can find some of these models? I'm trying to figure out if...
  35. R

    Is there a representation diagram for fermions like the 8 fold way for mesons?

    I was wondering if there was something similar to the 8 fold way representation used on mesons for the fermions?
  36. K

    QM - Position/Momentum representation problem

    Homework Statement Write down the time independant Schrodinger eqn in the momentum representation for a particle with mass m when the potential is given by V(x) = \frac{1}{2} \gamma x^2 Given that a possible solution is given by \Phi(p) = e^{\frac{-Bp^2}{2}} determine B and the...
  37. C

    Problem in constructing Matrix representation in |↑↓> basis

    If I want to derive the matrix representation for operator Q in the |S1=1/2 ,m1> |S2=1/2 ,m2 > basis, where |Si,mi> are common eigenstates of S2 , Si,z for the ith particle. And I do it in this way: <↑↑|Q|↑↑> <↑↑|Q|↑↓> <↑↓|Q|↓↑> <↑↑|Q|↓↓> <↑↓|Q|↑↑> <↑↓|Q|↑↓> <↑↓|Q|↓↑> <↑↓|Q|↓↓> ...
  38. K

    Series Representation for Function

    Homework Statement Fin the Taylor series about x = 0 for: f(x) = 1 / (1-x2)2 Homework Equations g(x) = 1 / (1-x2) The Attempt at a Solution Differentiating g(x), the series representation of g'(x) is Ʃn(x2)n-1 Since f(x) = g'(x)/2x f(x) = Ʃ(n/2)x2n-1 I'm pretty sure that's right...
  39. K

    Linear AlgebraMatrix Representation Problem

    Homework Statement I'm practicing for my finals this coming week and I'm confused about these 2 problems. Homework Equations The Attempt at a Solution For e). I followed my notes and came up with D=U-1AU..since its asking for some basis to standard basis. And once computed, its the diagonal...
  40. P

    Power Series Representation of xln(1+x^2)

    Write a power series representation of xln(1+x2) My first instinct was to attempt to take the second derivative and then find the summation and then integrate but that approach seemed to be a dead end. Basically, the x thrown in there confuses me and you can't split the function into two...
  41. A

    Matrix representation for transformation

    Homework Statement See my blog http://blog.yahoo.com/_HYUKNALCF4ULVTXY2JYSRAWDEA/articles/173228/index?bb=0 Homework Equations The Attempt at a Solution My attempt is also on the blog http://blog.yahoo.com/_HYUKNALCF4ULVTXY2JYSRAWDEA/articles/173228/index?bb=0
  42. T

    Find the length of the curve given by the parametric representation

    Find the length of the curve given by the parametric representation... Homework Statement Calculate the length of the curve given by the parametric representation r(t) = t2(cos t; sin t; cos 2t; sin 2t) for 1≤ t ≤+1: Homework Equations The Attempt at a Solution I know that...
  43. B

    Matrix Representation of Jₐ for j=1

    Homework Statement The raising and lowering angular momentum operators, J-hat(subscript +), J-hat(subscript -) are defined in terms of the Cartesian components J-hat(subscript x), J-hat(subscript y), J-hat(subscript z) of angular momentum J-hat by J-hat(+)=J-hat(x)+iJ-hat(y) and...
  44. S

    Can two functions have the same frequency domain representation?

    Hi, I stumbled upon the following two functions which have the same freq domain representation, 1/(j.pi.f) 1.signum(t) = u(t) - u(-t) 2. 2u(t) what is the reasoning behind them having the same f domain representation?
  45. A

    Geometric representation of two-forms.

    I've been browsing through MTW recently and I found something that puzzles me: They claim that if you have two form, call it \mathbf{T}, it's value, say \mathbf{T}(\mathbf{u} , \mathbf{v} ) can be represented geometrically as follows: take two vectors \mathbf{u} and \mathbf{v}; the surface...
  46. lpetrich

    Lie-algebra representation powers - plethysms

    I've done some more work on my http://homepage.mac.com/lpetrich/Science/SemisimpleLieAlgebras.zip package, adding decompositions of representation powers. It was only recently that I discovered that they are called "plethysms". Rep powers can be decomposed by symmetry types, a feature that can...
  47. U

    Can x²+2xsin(xy)+1=0 be solved for a single numerical solution?

    x²+2xsin(xy)+1=0
  48. B

    Why do q's form a complete commuting set of observables?

    I have a doubt from Dirac’s book which states that , Pr =-iħ∂/∂qr This equation was concluded after stating that the linear operator iħ∂/∂qr satisfy the same commutation relations with the q’s and with each other as p’s do. Next it is stated that(Dirac, p-92) “ This possibility enables us to...
  49. T

    Tensor products of representation - Weyl spinors and 4vectors

    Hi guys! I'm having some problems in understanding the direct products of representation in group theory. For example, take two right weyl spinors. We can then write\tau_{0\frac{1}{2}}\otimes\tau_{0\frac{1}{2}}=\tau_{00}\oplus\tau_{01} Now they make me see that...
  50. A

    Simple Q about direct product representation of a group

    (At least, I think it's simple.) Disclaimer: I'm approaching this subject from the vantage point of a chemist, so be careful with how much lingo/jargon/rigor you lay on me :redface: The claim is that if you have two representations of a group, \Gamma_1 and \Gamma_2, with bases \{ f_i \} and \{...
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