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. G

    Complex Analysis: Proving a function is equivalent to its series representation

    Homework Statement Compare the function f(z) = (pi/sin(pi*z))^2 to the summation of g(z) = 1/(z-n)^2 for n ranging from negative infinity to infinity. Show that their difference is 1) pole-free, i.e. analytic 2) of period 1 3) bounded in the strip 0 < x < 1 Conclude that they are...
  2. Q

    Matrix Representation of Nilpotent Linear Operator

    Homework Statement (5.5)V = Z_{\xi_{1}} \oplus Z_{\xi_{2}} \oplus ... \oplus Z_{\xi_{k}}, the basis for V is: \xi_{1}, \eta(\xi_{1}), ..., \eta(\xi_{1})^{p_{1} - 1} \xi_{2}, \eta(\xi_{2}), ..., \eta(\xi_{2})^{p_{2} - 1} . . . \xi_{k}, \eta(\xi_{k}), ..., \eta_(\xi_{k})^{p_{k} - 1}...
  3. P

    Parametric representation of a plane inside a cylinder

    Homework Statement find parametric representation for the part of the plane z=x+3 inside the cylinder x2+y2=1 The Attempt at a Solution intuitively... the cylinder is vertical with the z axis at its centre. and the plane is the whole surface inside the cylinder where y=0... visually...
  4. M

    How to Determine Validity of Power Series Representations for s(x) and s'(x)?

    Homework Statement Find the power series representation for s(x) and s`(x) integral sin (pi t^2)\2 and which of them is valid ? Homework Equations The Attempt at a Solution I tried to solve this question , but i am not sure s`(x) = sin (pi t^2)\2...
  5. tom.stoer

    SU(n) - conjugate representation

    Very simple question, but I can't find the answer. Taking an su(n) Lie algebra with hermitean generators we have [T^a, T^b] = if^{abc}T^c One immediately finds that the new generators \tilde{T}^a = (-T^a)^\ast define the same algebra, i.e. fulfil the same commutation relations...
  6. B

    Representation of a Matrix to a basis

    Homework Statement This problem refer to my previous post "trace of a matrix" M = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 1 & 5 \\ 0 & 5 & 3\end{pmatrix} from the following basis set: \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 &...
  7. B

    Calculating Representation of Linear Operator for Symmetric Matrix

    Homework Statement Let L(x) a linear operator defined by setting the diagonal elements of x to zero. What will be the representation of this operator to the following basis set? x E X. X denote the set of all real symmetric 3x3 matrices. Homework Equations L*y=x L=x*inv(y)...
  8. K

    Power series representation of ln((1+2t)/(1-2t))

    Homework Statement Find a power series representation for the function f(t) = \ln((1+2t)/(1-2t)) Homework Equations f(t) = \ln((1+2t)/(1-2t)) The Attempt at a Solution \ln(1+2t)-\ln(1-2t) take derivative of f(t) expanded \frac{2}{1+2t}+\frac{2}{1-2t} 2 \int \frac...
  9. K

    Power series representation question

    Homework Statement Determine the value of f(-1) when Homework Equations f(x) = (x/2^2) + ((2x^3)/2^4)+((3x^5)/2^6)+... . (Hint: differentiate the power series representation of ((x^2)-2^2)^(-1).) The Attempt at a Solution I was not very sure were to begin on this one. So I...
  10. P

    Representation of lie algebra of SL(2,C)

    Lie algebra \mathfrak{sl}(2,\mathbb{C}) consists of all 2x2 complex traceless matricies. The space of these matricies is 6-dimensional vector space over real numbers field but is 3-dimensional space over complex numbers field. Number of different representations of this algebra depend on how...
  11. A

    Representation of Angular Momentum Operator in the (j,j')

    Hello All, I'm trying to understand how the (j,j') representation of the Lorentz group. Following Ryder, I can see why we define A=J+iK and B=J-iK, which each form an SU(2) group. So it's clear to me what the rep of these generators is when acting on a state (j,j'): Rep(A)\otimes1+1\otimes...
  12. Y

    Question on TEM representation used in Griffiths book

    I am confused with this part of "Introduction to Electrodynamics" 3rd edition by Dave Griffiths. On page 367, the traveling wave is represented by: f(z,t)\;=\; A cos[k(z-vt) \;+\; \delta] 9.7 Where v is velocity and kv=\omega. This give: f(z,t)\;=\; A cos[kz \;-\;\omega t \;+\...
  13. I

    Matrix representation of a closed symmetric operator

    Homework Statement Show that a closed symmetric operator has a matrix representation. Homework Equations There are lots. I'm hoping somebody familiar with linear operators in Hilbert spaces is reading this! The Attempt at a Solution Hi, I'm trying to prove that a closed...
  14. P

    SO(10) 16 representation, decomposition & young tableau

    Homework Statement I've been looking at Paul Langacker's "Grand unified theories and proton decay" for a course on GUTs. I'm stuck with the 16 irrep of SO(10), particularly I don't understand how to prove the statement that the 16 decomposes as 5* + 10 + 1. I can see why it's useful physics...
  15. P

    Which Books Simplify Weights and Roots in Lie Algebra?

    Hi, I am currently studying Lie Algebra in Particle Physics' by Howard Georgi. I am finding the notes on Weights and Roots quite confusing. Can anyone suggest another book which explains this bit in a better fashion?
  16. C

    Question About Qubits and the Representation of the Bloch Sphere

    I have a general question about extracting information from measurement of a qubit. Theoretically a qubit in a superposition state contains an infinite amount of information, but when measured collapses to a definite state and result. My question is this: Is there a way to obtain a value from...
  17. P

    What are the possible dimensions of representation of SL(2,C)?

    Is it true that there are only two inequivalent two-dimensional representation of SL(2,C) group and they are responsible for Lorentz transformation of left and right Weyl spinor.
  18. T

    Quantum mechanics: Matrix representation of the momentum operator

    Homework Statement Is the matrix representation of the momentum operator a diagonal matrix? We know the momentum operator is a Hermitian operator Homework Equations A_nm = <psi_n | A | psi_m> The Attempt at a Solution I calced: A_nm = <psi_n|A|psi_m>=a_m <psi_n|psi_m> = a_m...
  19. M

    Representation of space time in GTR

    It is not a homework but a general question concerning the nature of our spacetime in physics. Perhaps the question will appear to be not relevant and perhaps it is not exactly the good place to post it here. Is there a difference between the two following way of doing: (a) in treating the...
  20. honestrosewater

    Prove this representation of naturals is a bijection

    Homework Statement Given s : \mathbb{N} \to \mathcal{P}(\mathbb{N}) s(n) = \{i | \exists (b_0,\ldots,b_k) \in \{0,1\}^{k+1} [n = \sum_j b_j2^j \,\wedge\, b_i = 1]\} show that s is a bijection between N and the finite subsets of N. In other words, if you express n as the sum of powers...
  21. P

    Representation of Delta Function

    Hopefully people are still prowling the forums this close to christmas :) I want to show that sin(ax)/x is a representation of a delta function in the limit a->infinty i.e 1) It equals 0 unless x=0 2) integrated from plus minus infinity it equals 1 and 3) multiplying by an arbitrary...
  22. D

    Power Series Representation of a Function Help

    [b]1. Homework Statement : Find the power series for the function f(x)=5/(7-x), centered at c=-2. [b]2. Homework Equations : a/(1-r) [b]3. The Attempt at a Solution : I know that I need to divide by seven to get (5/7)/(1-(x/7)) and then rewrite in the form the sum of (a)(r)^n. I tried adding 2...
  23. P

    Physical representation of irrational numbers

    My question relates to a specific example, namely the square root of two. If one forms a right isosceles triangle with the hypotenuse equal to 2 (be it metres, centimetres or whatever) then the other two sides must equal the square root of 2. But the square root of 2 is an irrational number. If...
  24. T

    Laurent series representation of f(z)=(z-1)/(z-2) at z=i

    Homework Statement Find the Laurent series representation for f(z)=(z-1)/(z-2) at z=i. Homework Equations NA The Attempt at a Solution I have taken multiple derivatives but I keep getting stuck at what to do after I find my representation of my nth derrivative. PLEASE HELP
  25. R

    Representation of a function with the natural logarithm

    I've been asked to express the inverse hyperbolic secant function arcsech in terms of the natural logarithm and am unsure as to where to begin in solving such a problem? could someone please point me in the right direction?
  26. M

    General Representation Formula for Harmonic Functions in the 2-D Case

    Homework Statement Derive the Representation Formula for Harmonic Functions (i.e. \nabla^{2}u = 0) Homework Equations See attached pdf The Attempt at a Solution See attached pdf Sorry about deferring to the pdf, but I already had everything typed up in LaTex previously, and I...
  27. L

    Lorentz transformations hae a representation on the fields - meaning?

    I've just read the statement "The Lorentz transformations have a representation on the fields" Can anyone explain the meaning of the word representation? I can't seem to get a satisfactory explanation anywhere and the notes don't go into much more detail on it.
  28. D

    Scalar Product of Momentum Eigenvectors in terms of Little Group Representation

    I'm trying to derive the equation for the scalar product of one particle momentum eigenvectors \Psi_{p,\sigma} ( p is the momentum eigenvalue and \sigma represents all other degrees of freedom), in terms of the little group of the Lorentz group with elements W that take the standard four...
  29. A

    Power Series Representation of a Function when a is a polynomial

    Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...
  30. N

    Position-Space Kinetic Energy Operator: Does Representation Matter?

    Hi This is actually a question regarding some formalism of QM, but I guess this is the place to ask it. Say we are looking at some kinetic energy operator T = T(r, ∇r), which has the form T = \sum\limits_{i,j} {T_{i,j} \left| \psi_i \right\rangle \left\langle \psi_j \right|} in...
  31. N

    QM: Operator in momentum representation

    Homework Statement Hi guys As we have discussed earlier, we can represent some operator in an arbitrary basis by the use of the 1-operator: T = \hat{1} T \hat{1} = \sum\limits_{\sigma_a,\sigma_b } {\left| {\psi _{\sigma_a} \left( {r_i } \right)} \right\rangle...
  32. K

    Complex Representation of Free Vibration

    Hi all I am struggling with going between various representations of vibrations in paticular the complex form. I am using Rao as my text btw so for a free vibration and making it simple no damping the euqation of motion is mx^{..} + kx = 0 with the general solution being x...
  33. srfriggen

    Linear Transformation; Geometric Representation

    Homework Statement (note; all column vectors will be represented as transposed row vectors, and matrices will be look like that on a Ti-83 or similar) L: R^3 -> R^2 is given by, L([x1, x2, x3]) = [2x1 + x2 - x3 x1 + 3x2 +2x3]* *Matrix Relevant...
  34. D

    Unique representation in graded modules

    In atiyah's book on commutative algebra page 106 it says that elements in graded modules can be written uniquely as a sum of homogeneous elements. More precisely: If A = \oplus^{\infty}_{n=0} A_n is a graded ring, and M = \oplus^{\infty}_{n=0} M_n is a graded A-module, then an element y \in...
  35. U

    Integral representation of modified Bessel function of the second kind

    Hi all. I need an integral representation of z^{-\nu}K_{\nu} of a particular form. For K_{1/2} it looks like this: z^{-\frac{1}{4}}K_{1/2}(\sqrt{z}) \propto \int_{0}^{\infty}dt\exp^{-zt-1/t}t^{-1/2} How do I generalize this for arbitrary \nu? A hint is enough, maybe there's a generating...
  36. P

    How Does the Dyson-Maleev Representation Benefit Solid State Physics?

    \hat{S}^+_i=\sqrt{2S}(\hat{a}_i-\frac{1}{2S}\hat{a}^+_i\hat{a}_i\hat{a}_i) \hat{S}^-_i=\sqrt{2S}\hat{a}^+_i, \quad \hat{S}^z_i=S-\hat{a}^+_i\hat{a}_i Why is in solid state physics often convenient to use this representation? It is obvious that (\hat{S}^-_i)^{\dagger}\neq \hat{S}^+_i...
  37. S

    Fundamental and Adjoint Representation of Gauge Groups

    Basic question, but nevertheless. In a non-Abelian gauge theory, the fermions transform in the fundamental representation, i.e. doublets for SU(2), triplets for SU(3), while the gauge fields transform in the adjoint representation, which can be taken straight from the structure constants of...
  38. N

    Find Matrix Representation of Green's Function | References

    How can I find the matrix representation for Green's function. If somebody have any reference please write it to me.
  39. D

    1 Dimensional Representation of a Gaussian Distribution

    Hi, I currently have a Gaussian distribution (Normalized Frequency on the y-axis and a value we can just call x on the x-axis). So for the sake of simplicity, let's say that I ignore any values below 0 and any values above 1 on the x-axis. Then what I will do is take 10 equal segments...
  40. F

    How Does Qubit Orientation on the Bloch Sphere Indicate Quantum States?

    There is something that I don't quite understand in relation to the Bloch Sphere representation of qubits. I've read that any vector on the sphere is a superposition of two basic states, like spin up and spin down, denoted by |1> and |0>. So does this mean that if the vector is at z=0...
  41. E

    Fourier series representation for F(t) = 0 or sin(wt) [depending on range]

    Homework Statement Obtain the Fourier series representing the function F(t)=0 if -2\pi/w<t<0 or F(t)=sin(wt) if 0<t<2\pi/w. Homework Equations We have, of course, the standard equations for the coefficients of a Fourier expansion...
  42. D

    Parametric representation of a Spiral

    Propose a parametric representation of a spiral. Hint: Use the parametric representation of a circle. This is the parametric representation of a circle we are given : x = r * Cos(Theta) y = r * Sin (Theta) 0 <= Theta <= 2 Pi Nope, we are not given anything background on spirals...
  43. G

    How to calculate parametric representation of a circle?

    Homework Statement y^2 + 4y + z^2 = 5, x = 3 Homework Equations The Attempt at a Solution I know that the calculated coordinates must satisfy the above equation, but I don't know how to go about solving for those coordinates. The best I could do was to equate z = \sqrt{(-y + 5)(y + 1)}.
  44. G

    Parametric representation of a straight line

    Homework Statement There are two questions, 1) straight line through (2, 0, 4) and (-3, 0, 9) 2) straight line y = 2x + 3, z = 7x Homework Equations r(t) = a + tb = [a1 + tb1, a2 + tb2, a3 + tb3] The book also explains how to calculate the line if b is a unit vector, but I don't...
  45. stevmg

    Derivation of Hyperbolic Representation from Lorentz/Minkowski equations in SR

    This is a carryover from a previous thread: https://www.physicsforums.com/showpost.php?p=2875138&postcount=68 Sports Fans: I am familiar with the Minkowski equations and the Lorentz transformations in one or two dimensions: A) In algebraic form (1) t2 - x2 = t'2 - x'2 (2) t' =...
  46. M

    Representation of covariant and contravariant vectors on spacetime diagrams

    Hi, How can we represent covariant and contravariant vectors on curved spacetime diagrams? How can we draw these vectors on a spacetime diagram? Contravariant vectors are really vectors, therefore we can represent them on the diagram with directed line elements. Covariant vectors are...
  47. J

    Graphical Representation of Cross Product

    Homework Statement Show graphically how \vec{a}\times\vec{x}=\vec{d} defines a line. \vec{a} and \vec{d} are constants. \vec{x} is a point on the line.Homework Equations \vec{a}\times\vec{x}=a\cdot x\cdot sin(\theta)\cdot \hat{n}The Attempt at a Solution Not sure if the included relevant...
  48. M

    SU(2) as representation of SO(3)

    The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))? Is a representation R of some group G a group too? If so, is it true that G is representation of R?
  49. Q

    Representation of Dirac-delta function

    Homework Statement Show that \delta_\epsilon(x) = \frac{\epsilon}{\pi (x^2 + \epsilon^2)} is a representation of the Dirac delta function. Homework Equations I already know how the function can satisfy the first two requirements of being a dirac delta function, namely...
  50. L

    Complex representation of a signal, quadrature signals in receivers

    (I posted this in the elctrical engineering forums because it's technically not homework, but it probably belongs here...it got no replies there) 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...
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