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. D

    About the operatior-sum representation

    Homework Statement Homework Equations Suppose we have a single qubit principal system ,interacting with a single qubit environment through the transformation U=P0\otimesI+P1\otimesX where X is the usual Pauli matrix (acting on the enviornment)and P0=|0><0| ,P1=|1><1| are projectors...
  2. C

    Can someone help me find the power series representation for this function?

    I'm trying to do the question attached. I got the first three answers correct knowing that the nth derivative of a function evaluated at 0 divided by n! = c_n. However, I did the same for the others and the answer is incorrect. I know that I need the power series representation of that function...
  3. S

    Finding a power series representation

    Homework Statement Start with the power series representation 1/(1-x) = sum from n=0 to inf. of x^n for abs(x) < 1 to find a power series representation for f(x) and determine the radius of convergence. f(x)=ln(5+x^2) Homework Equations The Attempt at a Solution Okay, so I...
  4. U

    What is the principal representation?

    Question on representation theory. What is the principal representation? I would like a good clear definition. I can't find it in my book (bad index) nor can I find it on the web.
  5. M

    Matrix representation for (S1+S2)^2 operator

    Hi o:) I wasn't able to insert equations here, so here's the link where my problem is described: http://www.advancedphysics.org/forum/showthread.php?t=7696
  6. K

    Power Series Representation

    Homework Statement Find a power series representation for the function f(x) = x / (4+x) and determine the interval of convergence. I have no idea how to begin this problem. My only guess would be trying to divide something out in order to simplify to something that I'm able to...
  7. A

    Some questions on representation theory

    Hello all, Can anybody help me solve the following exercises from the book: Representation Theory, A first course by William Fulton and Joe Harris,1991 page 138, exercise 10.3 page 140, exercise 10.7 page 141, exercise 10.9 Thanks in Advance,
  8. K

    Mathematica Mathematica Representation of a wave

    Homework Statement "A sinusoidal wave is propagating along a stretched stringas a function of time is graphed in the figure (attached) for particles at x = 0 and x =0.0900m. a) what is the period of the wave ( my ans = 0.04 sec b) what is the amplitude of the wave ( my ans = 4mm)...
  9. M

    Fourier representation of aperiodic irregular function

    We have \epsilon(i f, r) = \epsilon_2(i f) when H + h_2(x) \leq z < + \infty \epsilon(i f, r) = 0 when h_1(x) < z < H + h_2(x) \epsilon(i f, r) = \epsilon_1(i f) when - \infty < z \leq h_1(x) show the corresponding Fourier transform is \frac{i}{q_z} \int d^2x e^{iq_\bot \cdot...
  10. M

    Fourier representation of CT peridic signals

    I want to find the Fourier coefficients for the following signal: \cos(2 \pi t)^2 Can I simply use the identity?: \frac{1}{2} + \frac{\cos(2 \pi t)}{2} And then use the complex definition: \frac{1}{2} + \frac{1}{4} (\exp{j2 \pi t} + \exp{-j2 \pi t}) From the synthesis equation I can...
  11. E

    Matrix representation of ladder operators

    Homework Statement Find the matrices which represent the following ladder operators a+,a_, and a+a- All of these operators are supposed to operate on Hilbert space, and be represented by m*n matrices. Homework Equations a+=1/square root(2hmw)*(-ip+mwx) a_=1/square root(2hmw)*(ip+mwx)...
  12. P

    Matrix Representation of a Linear Map

    Homework Statement Consider the linear map A : R3 ----> R3 given by A(x1, x2, x3) = (x1 − x2,−x1 + x2, x3). (a) Find the adjoint map A^*. (b) Obtain the matrix representations of A and A* with respect to the canonical basis f_1 = [1, 2, 1], f_2 = [1, 3, 2], f_3 = [0, 1, 2]...
  13. Q

    Standard representation of a vector space

    Hi everyone, Can anyone explain the following to me? Given a basis beta for an n-dimensional vector space V over the field F, "the standard representation of V with respect to beta is the function phi_beta(x)=[x]_beta for each x in V." This is from my textbook. It then proceeds to give...
  14. M

    Decimal representation of real numbers

    I'm doing self study out of Apostol's Calculus vol. I and I got stuck trying to prove what the author writes is easy to verify, but I can't get my head around it. Basically, this is the problem statement from page 31, last paragraph: Given a positive real number x, let a0 denote the largest...
  15. E

    Express for Operator of coordinate in momentum representation

    Homework Statement 1. Obtain an expression for operator of coordinate in momentum representation. To this end begin with definition of the average coordinate x = ∫ψ*(x)xψ(x)dx express the wave functions as wave packets in terms of plane waves, and rewrite the expression for average...
  16. Q

    Finding Unique Vector Representation

    unique vector representation?? Homework Statement So, here I am again with another... The problem gives me a basis {(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1)} in F^4. I am supposed to find the unique representation of an arbitrary vector (a1,a2,a3,a4) as a linear combination of the vectors...
  17. R

    Quick Binary Representation Help

    Homework Statement What is the range of a 11bit code, using c. Unsigned fixed-point with 8 binary places Homework Equations Base conversions 2^n where n is bits, for range/precision The Attempt at a Solution I have two ideas: 1) The highest value that can be represented is...
  18. P

    Matrix representation

    I've given the linear transformation L: \mathbb{R}^4 \rightarrow \mathbb{R}^3 , where L(\mathbf x) = A\mathbf x and A = \left[ \begin{array}{cccc} 1&1&-4&-1 \\ -2&3&1&0 \\ 0&1&-2&2 \end{array} \right] The first question of the problem is that, by using the standard basis E for...
  19. G

    How can I solve the parametric representation problem for x^3+y^3=u^3+v^3?

    Does anyone have any ideas on how to even start this problem? I am supposed to find a general solution in rational numbers for (aside from the trivial ones): x^3+y^3=u^3+v^3 Actually, I'm given the answer (which is really messy) and am supposed to show how to derive it. The book gives the...
  20. B

    Help with power series representation

    Homework Statement find a power series representation for the function and determine the radius of convergence. f(t)= ln(2-t) Homework Equations The Attempt at a Solution i first took the derivative of ln(2-t) which is 1/(t-2) then i tried to write the integral 1/(t-2)...
  21. M

    MATLAB Complex Number Representation in Matlab

    Hey Everyone, I cannot seem to find an way in Matlab to convert a number which has a real and imaginary part in cartesian form into polar form and then express the polar representation on the output. Ex. Convert z=0.1602932442+0.8277219859*j Into 0.8431<79.04 deg (without using...
  22. S

    Matrix geometric representation

    Hello all, I have attached a matrix. I am trying to work out this matrix will transform a set of co-ordinates (x_1, y_1, z_1) to a new set of co-ordinates (x_2, y_2, z_2) Can anyone give me any hints on how to tackle this problem?
  23. S

    Maple Floating point representation in Maple

    How do i show the floating point reprensetation of a number like say ... 100! or 1000! in maple?? i am relatively new to maple so i am not familiar with many of the commands
  24. B

    Representation Algebra: Unanswered Equations

    (If the following equations do not display OK, try refreshing it a few times.) 3 \otimes 3 &=& \bar 3 \oplus 6, \quad qq, 3 \otimes \bar 3 &=& 1 \oplus 8, \quad q\bar q, \, \quad q[qq]_{\bar 3}, \bar 3 \otimes \bar 3 &=& ?, \quad \bar q\bar q, 3 \otimes 3 \otimes 3 &=& \left( \bar 3...
  25. D

    What Determines the Independent Variable in Different Quantum Representations?

    What's the relationship between an certain represetation and the independent variable under this represetation? for details: 1 Must the independent variable of X-represetation be X,and P-represetation must be p?if so ,What's the represetation which basic fuctions are P-latent fuctions...
  26. A

    Geometrical representation of the nth derivative

    The first derivative represents the slope of a tangent at a point on the function's curve. The second derivative represents the concavity of the function's curve. However I am unable to figure out what the other derivatives of a function represent either physically or geometrically. Pleae help.
  27. E

    Graphical Representation of v(t) Function

    I'm trying to represent graphically the following function: v(t)=3 + 1.414cos(w0t) - 1.414sin(w0t) + 2cos(2w0t + 5pi/2) The problem is that I'm not sure how I represent the sine and cosine both in the same graph...I know that 3 is the continuous component and the w0t equals the "x"...
  28. S

    QFT and unitary Lorent representation

    Hey guys! A question: My QFT Lagrangian, fournishes through Noether's thm plus relativistic invariance a supposedly unitary and a supposed representation of the Lorentz group. These are the operators meant to act on my Hilbert space of possible states. What guarantees that this actually...
  29. E

    Is there an infinite product representation of e^(z)

    I was wondering if anyone knew where I might find, or formulate, the infinite product representation of the entire function f(z) = e^(z). Wikipedia says, and I qoute, "One important result concerning infinite products is that every entire function f(z) (i.e., every function that is...
  30. M

    Fourier representation of fields

    Hello, From Jean Zinn-Justin: "The conducting plates impose to the electric field to be perpendicular to the plates. It is easy to verify that this condition is satisfied if the vector field A_\mu itself vanishes on the plates. Calling L the distance between the plates, z=0 and z=L the...
  31. M

    Anyone prove Fourier representation of the Coulomb potential

    I've seen the Fourier representation of the Coulomb potential is -\frac {Ze} {|\mathbf{x}|} = -Ze 4\pi \int \frac {d^3q} {(2\pi)^3} \frac {1} { |\mathbf{q}|^2} e^{i\mathbf{q}\cdot\mathbf{x}} Will anyone show me how to prove it? (yes, it's the Coulomb potential around an atomic nucleus.)...
  32. Oxymoron

    How Does Positivity in C*-Algebras Relate to Their Representations?

    If I have a faithful nondegenerate representation of a C*-algebra, A: \pi\,:\,A \rightarrow B(\mathcal{H}) where B(\mathcal{H}) is the set of all bounded linear operators on a Hilbert space. And just suppose that a\geq 0 \in A. How is the fact that a is positive got anything to do with...
  33. P

    Dual Representation: Why Use g^{-1}?

    I've been starting to study some things about representation theory. I've come to the point where they introduced the dual of a representation. Suppose that \rho is a representation on a vector space V. They then define the dual representation \rho^* as: \rho^*(g) = \rho(g^{-1})^t: V^*...
  34. MathematicalPhysicist

    Mathematica Mathematical representation and its implication in physics.

    my post i really about discussing the difference that a mathematical representation does to physics. for example, let's take for example a system which can be be represented by quaternions and by complex numbers. if we can represent a physical theory by quaternions which aren't commutative...
  35. 0

    Set Representation: Convey Data Without Wasting

    Is there a general good way to represent sets as sequences, without wasting data or conveying extra data? One way is a membership list like 11010100111010 but this only works if the possible number of elements is small. I am thinking, data _other_ than ordering, directly about the set objects...
  36. G

    Find the fourier series representation of x^2

    ok, i wasn't sure if i ought put this in math or phys, we're going over it my phys class, but its just math... whatever.. So i had to find the Fourier series representation of x^2 in the intervals (-pi, pi) and (0, 2pi). i haven't even started the (0, 2pi) one, cause i can't get the first...
  37. E

    Unit Vector to Verbal Representation

    I'm currently taking my first physics class, and on the very first day, my teacher began discussing vectors as if everyone knew what Unit Vectors were. I've managed to understand most of his lecture thus far, but I still don't understand- How do you convert from Unit vectors (denoted by i...
  38. L

    Källen-Lehmann Representation

    I have two basic questions about the full propagator (2-point function) in QFT. Am I correct that for a scalar field, it is \frac{iZ}{p^{2}-m^{2}+i \epsilon} + \int_{m^{2}}^{\infty} dX \frac{\rho[X]}{p^{2}-X+i \epsilon} ? (1) Is this form of the propagator a feature of _quantum_ field theory...
  39. T

    Finding Power Series Representation for arctan(x): Help Needed!

    may i know how to solve this ques:find the power series representation for arctan (x) i know that arctan (x) = integ 1/(1 + x^2) but then from here i don't know how to continue. pls help...
  40. dextercioby

    Representation theory on RHS

    Has it been done...? If so, any textbook on it...? The concept of Rigged Hilbert Spaces is essential to Quantum Mechanics. Symmetries are implemented in physics by doing representation theory of some symmetry groups, almost all of them being Lie groups: Galilei group, Poincaré group...
  41. M

    Qestion: Vector field and (n-1)-form representation of current density

    Question: Vector field and (n-1)-form representation of current density Electric current density can be represented by both a vector field and by a 2-form. Integrating them on a given surface must lead the same result. My question is, what is the relation between this vector field and the...
  42. E

    Finding the Interval for Theta in Parametric Representation of a Sphere

    Find the parametric representation for the surface: The part of the sphere x^2 + y^2 + z^2 = 16 that lies between the planes z = -2 and z = 2. okay, i know that i have to use spherical coordinates which is x = 4sin(phi)cos(theta) y = 4sin(phi)sin(phi) z = 4cos(phi) i know how to find...
  43. I

    Parametric Representation of Field Lines

    F(x,y,z)=(-\frac{y^2+2z^2}{x^2},\frac{2y}{x},\frac{4z}{x}) "Find parametric representations of the field lines." How do I parametrize all possible field lines?
  44. M

    What is Group Theory and How Does it Relate to Representation Theory?

    I think I will write out the beginnings of a "book" that I'd been meaning to get around to starting at some point. Better that someone might actually read it here than trust to them finding it on my web page. Any requests, let me know by PM (or if this is inappropriate for this forum)...
  45. kakarukeys

    Finite Dimensional Representation of SU(2)

    Does anybody know whether the following irreducible representations of SU(2) are unitary? g belongs to SU(2) [U_j(g) f](v) = f(g^{-1} v) f is an order-2j homogeneous complex polynomial of two complex variables v = (x, y) e.g. for j = 1, f = 2x^2 + 3xy + 4y^2
  46. R

    What a representation of a group is please?

    Can please explain to me what a representation of a group is please? Hopefully something more illuminating than what I might find on wikipedia or eric weissteins mathworld. Also, perhaps illustrated with an example.
  47. R

    What is a 'representation' in Field Theory

    I see this term a lot. I know its something to do with a group but I'm not too sure what it is. Also - how does a representation (mathematical concept) translate into particle physics concept.
  48. Norman

    Dual Representation and anti-particles

    I am reading ahead for my Group Theory introduction to QFT and I have a question about the dual representation. If the dual representation is the same as the ordinary representation, that is to say the ordinary representation is "real", how do we represent anti-particles in this case? This...
  49. N

    Representation of Rational Numbers by Formulas

    Dear Experts. Do you know some literature on representation of rational numbers by formulas? TIA.
  50. P

    Understanding Linear Vector Spaces for System Representation

    what is the reason behind choosing the linear vector spaces in representing the state of a system? why is it convenient ? and why do we actually need a linearity ?
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