Representation Definition and 766 Threads

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. B

    Proving Equivalence of Euler-Macheroni Constant

    Hi Everyone, I just registered for PF today because this problem was driving me nuts and I was hoping to get some help. It comes from pg. 5 of Peter Miller's "Applied Asymptotic Analysis" and goes like this: The Euler gamma constant has one definition as \gamma := \int_0^\infty...
  2. L

    Complex representation of a signal, quadrature signals in receivers

    Hey, I'm hoping this thread can clear up some confusion I have with complex signals and moving back and forth from physical signals to the mathematical models. I'll probably ask some questions specifically, but if you would like to help me please treat this whole post as a question because I'll...
  3. jegues

    Parametric representation of curves

    Homework Statement See figure. Homework Equations N/A The Attempt at a Solution I've dealt with parametric equations for lines before in my linear algebra class but I'm not sure how I'm suppose to model two curves with one. Anyone have any suggestions?
  4. G

    Limit question series representation

    Homework Statement Lim (t->0+) [exp(-x^2/t)] / sqrt(t) The Attempt at a Solution Ive looked at series representation: 1/sqrt(t) * [exp(-x^2) - 1+t^2/2 + t^3/6 + ...] but that doesn't help. L'Hopital's rule isn't any use either because the t terms are in the denominator but I know...
  5. K

    Representation of angular momentum matrix in Cartesian and spherical basis

    The two sets of matrices: {G_1} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ \end{array}} \right){\rm{ }}{G_2} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 1 \\ 0 & 0 & 0 \\ { - 1} & 0 & 0 \\ \end{array}} \right){\rm{...
  6. N

    Irreducible representation of so(3)

    Hi guys, I have a question which is very fundamental to representation theory. What I am wondering is that whether a first rank cartesian representation of so(3) is irreducible. As I understand first rank cartesian representation of so(3) can be parametrized in terms of the Euler angles. That...
  7. B

    Arbitrary function of a matrix - power-series representation

    Homework Statement I am trying to evaluate exp(i*f(A)), where A is a matrix whose eigenvalues are known and real. Homework Equations You can expand functions of a matrix in a power-series. I think that's the way to get started on this problem. I foresee the exponential of a power-series...
  8. T

    Rational Representation of a Repeating Decimal

    Homework Statement Find the rational number representation of the repeating decimal. 1.0.\overline{36}Homework Equations The Attempt at a Solution I know it has something to do with infinite geometric sequences but I'm not sure what. what would your ratio be for a repeating decimal, I've...
  9. M

    Power Series Representation for Arctan(x)

    Homework Statement F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)? en=? an=? pn=? I know en = n-1 Homework Equations The Attempt at a Solution Start with the geometric series 1/(1 - t) = ∑(n=0 to ∞) t^n. Let t = -x^2: 1/(1 + x^2) = ∑(n=0 to ∞)...
  10. L

    Hilbert Spaces for PDEs: How are they used?

    Hi, When I solve the diffusion equation for a spherically symmetric geometry in spherical coordinates I obtain the following general solution (after application of the boundary conditions). T(r,t) = \sum_{n=1}^{\infty}\, \frac{A_n}{r}\sin(\lambda_nr)\exp(-\alpha\lambda_n^2t) So to...
  11. R

    Graphical representation of complex roots to equations

    I've never properly studied complex numbers but I will soon (in September). Basically: We get taught from a young age that: the real root of f(x)=x²-4 is where the graph of y=f(x) cuts the x axis But is there a graphical representation of a complex root? What's so special about the...
  12. A

    Proving Periodicity of Sin/Cos with Series Representation

    with the series representation of sin or cos as a starting point (you don't know nothing else about those functions), how to prove: a. they are periodic. b. the value of the period.
  13. Z

    A question on proof of Riesz Representation Theorem when p=1

    This question comes from the proof of Riesz Representation Theorem in Bartle's "The Elements of Integration and Lebesgue Measure", page 90-91, as the image below shows. http://i3.6.cn/cvbnm/ac/9a/a3/3d06837bc78f74ba103b6d242a78e3a1.png The equation (8.10) is G(f)=\int fgd\mu. The...
  14. P

    How to Draw Incident and Reflected Waves for Barrier Potential?

    Having trouble drawing the wave representations for this diagram. [PLAIN]http://img156.imageshack.us/img156/94/diag.png I'm not sure what the incident and reflected waves should look like. The question is asking to draw the waves in each region. I understand how to calculate the waves just...
  15. N

    Is this a correct taylor series representation centered at 1

    f(x)=1/(1-x^2)^(1/2) 1/x^(1/2)=1+ sum(( (-1)^n 1*3*5*7...(2n-1)(x-1)^n )/(2^n n! ) , n=1, infty ) thus 1/(1-x^2)^(1/2) = 1+ sum(( 1*3*5*7...(2n-1)(x^2)^n )/(2^n n! ) , n=1, infty ) is this a correct taylor series representation centered at 1
  16. Z

    Is the hamiltonian in the coordinate representation always diagonal?

    In the coordinate representation of a quantum mechanical system, is it always true that the Hamiltonian of the system is diagonal? If so, can someone explain to me why this is true?
  17. N

    A basic question on representation theory

    Homework Statement Hey, this is problem 4A out of Georgi (Lie Algebras in particle physics). The question says given an operator O_x , x=1,2, in the spin 1/2 rep of SU(2). [J_a,O_x]=O_y(\sigma_a)_{yx}/2 where \sigma_a are the Pauli spin matrices. Given A=\langle 3/2, -1/2...
  18. N

    Converting Power Series to Integrals: How to Handle Constants of Integration?

    \int \frac{x-arctanx}{x^3}dx \frac{d}{dx}( x-arctanx ) = 1-\frac{1}{1+x^2}=\frac{x^2}{x^2+1} = x^2 \sum_{n=0}^{\infty}(-1)^nx^{2n} = \sum_{n=0}^{\infty}(-1)^nx^{2n+2} \int \sum_{n=0}^{\infty}(-1)^nx^{2n+2} dx = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}+C C=0? \int...
  19. M

    Find a Fourier Series representation

    I'm having problem finding the representation for the Fourier series with function f of period P = 2*pi such that f (x) = cosαx, −pi ≤ x ≤ pi , and α ≠ 0,±1,±2,±3,K is a constant. Any help is appreciated...
  20. T

    Solving differential equation using power series representation

    Homework Statement The problem is: (x2 - 4) y′′ + 3xy′ + y = 0, y(0) = 4, y′(0) = 1 Homework Equations Existence of power series: y = \sum c(x-x0)^n or y = (x-x0)^r\sum c(x-x0)^n The Attempt at a Solution I know the point x=2 is an ordinary point of the differential...
  21. Oddbio

    Different representation of Laplacian

    I am trying to show that the laplacian: L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} can also be represented as: L = \frac{1}{2}(\vec{E}^{2}-\vec{B}^{2}) where F^{\mu\nu} = \partial{}^{\mu}A^{\nu} - \partial{}^{\nu}A^{\mu} and F_{\mu\nu} = g_{\mu\alpha}F^{\alpha\beta}g_{\beta\nu} A is the...
  22. S

    Power Series Representation of ln(1+7x)

    how can i find a power series representation for a function like f(x)= ln(1+7x)?
  23. S

    Finding Power Series Representation of Derivatives: 1/x-9

    how can i find a power series representaion of d/dx (1/x-9)
  24. M

    Gamma function (infinite product representation)

    I have come across this expression in some notes \Gamma (z) = \frac{1}{z} \prod \frac{(1+ \frac{1}{n})^{z}}{1+ \frac{z}{n}} Do you think it's accurate? I have some doubts because I have looked for it on wokipedia, and I couldn't find it.
  25. I

    Representation of properties of Complex Numbers in Argand Diagrams

    Homework Statement Draw an argand diagram to represent the follwing property: real(z) < abs(z) < real(z)+img(z) Homework Equations z = x+iy; real(z) = x abs(z) = sqrt(x^2 + y^2) img(z) = y The Attempt at a Solution substituting original expression with x, y, and sqrt(x^2 + y^2)...
  26. L

    Book recommendation for representation theory(physicist)

    Hi Could anyone recommend me a good book that will teach me the kind of group/representation theory I would need to understand these things when applied to QFT (Lie Algebra, Lorentz group, SU(2) etc)? Thanks
  27. F

    Finding Chen's Paper: "On the Representation of a Large Even Integer

    Hi guys, Not actually a mathematics question as such (sorry) but does anyone know where i can get my hands on a copy of Chen's paper "On the representation of a large even integer as the sum of a prime and the product of at most two primes". For the life of me all i can find is references to it...
  28. I

    What is the explicit representation of 2cos(omega)t - 1/4 sin(omega)t = 0?

    explicit representation?? I got to show the explicit representation of 2cos(omega)t - 1/4 sin(omega)t = 0. what is this? is this operator notation??
  29. M

    Fourier Series Representation Problem

    Homework Statement Since I don't know how to insert equations into a message here, I've scanned both the problem and my attempt at a solution. Where I run into problems is how to find an. I'm not completely sure how to treat that integral and was hoping somebody could nudge me in the...
  30. Z

    Representation by a diagonal matrix question

    Homework Statement Let T be the linear operator on R3 that has the given matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. a) Determine whether T can be represented by a diagonal matrix, and b) whenever possible, find a diagonal matrix and a basis of R3 such that T is represented...
  31. D

    Feynman propagator on the cylinder - position space representation

    Hi all! Does anyone know the position space representation of the Feynman propagator on the cylinder? The momentum space representation is the same as in Minkowski 2D space, but the position space representation is different because the integrals over momenta are now sums. Or could someone...
  32. B

    Questions about a projection operator in the representation theoy of groups

    D(g) is a representaiton of a group denoted by g. The representaion is recucible if it has an invariant subspace, which means that the action of any D(g) on any vector in the subspace is still in the subspace. In terms of a projection operator P onto the subspace this condition can be written...
  33. quasar987

    Representation of finite group question

    Does anyone know how to prove that any irreducible representation of a finite group G has degree at most |G|? Equivalently, that every representation of degree >|G| is reductible. Thx!
  34. F

    Basic matrix representation question

    Hi guys, Just brushing up on my GR for a project and i have a silly question: For the spherical polar representation of the schwarzschild metric, the fact that there are no infintesimal-cross terms implies that the non-diagonal entries in the matrix representation are zero, correct? I...
  35. S

    Can Two Different Base Representations of an Integer Prove It is Unique?

    base representation please help~ It is known that if asks+as-1ks-1+...+a0 is a representation of n to the base k, then 0<n<=ks+1-1. Now suppose n=asks+as-1ks-1+...+a0 and m=btkt+bt-1kt-1+...+b0 with as,bt not equal to 0, are two different representations of n and m to base k, respectively...
  36. S

    Proving the Uniqueness of Base Representation for Integers

    base representation please help~ It is known that if asks+as-1ks-1+...+a0 is a representation of n to the base k, then 0<n<=ks+1-1. Now suppose n=asks+as-1ks-1+...+a0 and m=btkt+bt-1kt-1+...+b0 with as,bt not equal to 0, are two different representations of n and m to base k, respectively...
  37. A

    Transformation of the adjoint representation

    Homework Statement given that N\otimes\bar{N} = 1 \oplus A consinder the SU(2) subgroup of SU(N), that acts on the two first components of the fundamental representation N of SU(N). Under this SU(2) subgroup, the repsentation N of SU(N) transforms as 2 \oplus (N-2) with info...
  38. S

    Symmetry and irreducible representation

    1. could anyone give sort of a qualititative explanation of how symmetry and irreducible representation are related in the context of molecular spectroscopy? like why is it so useful to count how many symmetries a molecule has and what does it have to do with irreducible represenations and...
  39. S

    Irreducible representation of S3

    okay so i was reading a book on representations and found this discussion and was confused: http://books.google.com/books?id=Hm-aKMkKXzEC&lpg=PP1&dq=group%20theory%20and%20physics&pg=PA53#v=onepage&q=&f=false it starts at the bottom of pg 53 and ends at the top of pg 54 so I understood the...
  40. pellman

    Delta function representation from EM theory

    Claim: \nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x}) Anyone know of a proof of this? (or a reference which covers it?) We need to show that \frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0). The claimed identity can be seen in the solution for...
  41. V

    Matrix representation of a group

    I am unable to grasp how to generalise the concept of matrix representation for a group say like D3 or D4.I know how to manipulate the 2x2 matrix,but how do I obtain a group in say a 3x3 matrix to show rotation and reflection?
  42. S

    A question on irreducible representation

    I'm having trouble grasping what an irreducible representation is. Can someone explain what this is through an example using SU(2) and/or SO(3) w/o invoking the use of characteristic? Like I'm reading a bunch of stuff but I'm not catching the significance of any of it. ...Imagine you were...
  43. pellman

    Complex integral representation of Dirac delta function?

    We all know that \frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x'). i am working a problem which appears to depend on the statement \int e^{z^*(z-w)}dz^*\propto\delta(z-w) Does anyone know if this is valid? \delta(z-w) is defined in the usual way so that...
  44. P

    Angular momentum and group representation

    I heard that angular momentum operators and their eigenvectors are realted to SO(3) or SU(2) group. Does anyone know a good textbook which explain the connection between how group theory and quantum mechanics (especially angualr momentum). I'm interested rather in books which emphasizes...
  45. R

    What is the power series representation for F(x)= 3/4x^3-5, where c=1?

    [b]1. Find a power series for F(x)= 3/4x^3-5, where c=1 [b]2. power series = 1/a-r [b]3. What I did was take a derivative to get a similar function that was easier to solve. I used 1/x^3-1. Then I found a series for that function. Which I got \sum(x^3)^n. Then I added back from my...
  46. F

    Riesz representation theorem example

    My current understanding of the Riesz representation theorem is that it is useful since it tells you what all bounded linear functionals on Lp look like. They look like the integral of fg where g is some function in Lq. So, I was trying to think of an example of a bounded linear functional on an...
  47. J

    Can you check my work? power series representation

    can you check my work? "power series representation" is ok I figure it out.
  48. M

    Confusion regarding to polar form representation of AC quantity

    I'm sure my question is very simple to most of u guys. But I have the following confusion. Let's say we have an AC voltage source in a circuit. In rectangular form it's phasor form is v= -4 - 16 j . I want to write this phasor in polar form. Well, The phasor is in 3rd quadrant of complex...
  49. Q

    Power series representation of a function

    Homework Statement Find a power series representation for f(x) using termwise integration, where f(x) = \int_{0}^{x} sin(t^3) dt . Homework Equations The Attempt at a Solution I've never done this before, but apparently, if I have a power series representation for sin(t^3), I...
  50. R

    Pauli lubanski pseudo vector in spin representation

    I'm trying to calculate the pauli-lubanski pseudo vector for different representations of the poincare group. The first rep is the infinite dimensional "angular momentum" rep where the generators of the lorentz part take the form : M_ab = x_a*d_b - x_b*d_a (for 3 rotations) M_ab =...
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