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.
Hi,
I'm currently reading a book on particle physics, which tells me this about SU(3):
"...The generators may be taken to be any 3x3-1=8 linearly independent traceless matrices. Since it possible to have only two of these traceless matrices diagonal, this is the maximum number of commuting...
Homework Statement
I want to find a matrix representation of the grassman algebra {1,x,x*,x*x} (and linear combinations with complex coefficients)
defined by [x,x]+=[x,x*]+=[x*,x*]+=0
I really don't know how to make matrix representations of an algebra. Is any set of 4 matrices that obey the...
Homework Statement
Express the surface
x = 2cos(theta)sin(phi) y=3sin(theta)sin(phi) z=2cos(phi)
as a level surface f(x,y,z) = 144,
f(x,y,z) = ?
Homework Equations
The Attempt at a Solution
I figured they wanted the equation f(x,y,z) in x^2+y^2+z^2=144 so I though that by...
Can a generic, not necessarily harmonic, signal of time be represented as a complex signal with a real and imaginary part?
Usually the complex rappresentation is used for time harmonic signals and linear systems.
The "real" time signal is transformed into a complex signal. At the end of the...
In the appendix B of Goldstein's classical mechanics (3rd edition), the authors discussed the dihedral group and said:
"Notice how the group elements in class 3 involve only \sigma_1 and \sigma_3. Thus, they are independent of the matrices I and \sigma_2, as is expected from the structure of...
Does anyone know of a series representation for:
\frac{sin(x)}{cos(x)+cosh(x)}
Preferably valid for 0<x, but any ideas or assistance on any domain would be much appreciated.
For some reason, the momentum representation in QM wasn't covered in our class, so I'm figuring it out on my own (no, this isn't homework...it's just me reviewing physics for the PGRE).
My question: what is the energy (kinetic energy, I guess) of a particle in the momentum representation of...
Hello, I am attempting a past exam paper in preparation for an upcoming exam. The past exam papers do not come with answers and I'm a little unsure as to whether I'm doing all of the questions correctly and would like some feedback if I'm going wrong somewhere.
Any help is greatly appreciated...
Homework Statement
Here it is: a particle in 1-d infinite potential well starts in state \Psi(x,0) = A Sin^{3}(\pi*x/a): 0\leqx\leqa.
Express \Psi(x,0) as a superposition in the basis of the solutions of the time independent schrodinger eq for this system, \phi_{n}(x) = (2/a)^{1/2}...
Some years ago I used the device of representing composite numbers by rectangular forms to demonstrate the structure of numbers to third grade students. Primes were represented by lines of various lengths. Number 10 would be a 2x5 rectangle and 20 a 2x2x5 rectangular solid. (I used various...
I have only very little knowledge of group theory and representation theory. There is a lot to learn, so I wondered what are the final contributions to physical understanding from representation theory?
I'd like to find some less abstract results. For example that the mass is the Casimir...
Homework Statement
\psi(x)=\frac{3}{5}\chi_{1}(x)+\frac{4}{5}\chi_{3}(x)
Both \chi_{1}(x) \chi_{3}(x) are normalized energy eigenfunctions of the ground and second excited states respectivley. I need to find the 'wavefunction in the energy representation'
The Attempt at a Solution...
Homework Statement
Determine a parametric representation for the part of the sphere x2 + y2 + z2 = 4 that lies between the planes z = 1 & z = -1.
Homework Equations
The Attempt at a Solution
We never learned spherical coordinates in class so I am not sure if I am using this...
Question in the title, ie why is Tr(T_{a_1}T_{a_2}...T_{a_n}) independent of which representation we choose, where the Ts are a matrix representation of the group generators.
What's the use of it? Anyone show a simple but illustrative example of the usefulness of representation theory?
I can see how faithful representations might be useful but not fully. What I can't imagine is how unfaithful representations can be of any use.
Thanks
I will soon move to Boston after living in Florida and Texas, Respectively. I was a little perturbed that I will have to pay state taxes. This was, until I found out the public school system in MA was ranked #1 and Texas and Florida #33 and #39, respectively [1]. I seem to me that the money is...
Hello everyone,
This is a normal modes problem that I’m working on, where the details are a bit tedious, but what I need to do is to write the following system:
mx`` = –2kx + ky + c
my`` = kx – ky + c
In the following form:
| m 0 | |x``| = |–2k k| |x|
| 0 m | |y``| =...
Homework Statement
Find the generators of the four dimensional irreducible representation of SO(3), such that J_3 is diagonal.
The Attempt at a Solution
I know how to get the rest if I know J_3, by using ladder operators. But what is J_3?
For a 3d representation it's diagonal with 1,0-1, in 4d...
I'm trying to understand this paper which the author claimed that he had constructed an infinite dimensional representation of the su(2) algebra. The hermitian generators are given by
J_x=\frac{i}{2}(\sqrt{N+1}a-a^\dagger\sqrt{N+1} )
J_y=-\frac{i}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )...
I'm trying to understand this paper on the representation of SU(2).
I know these definitions:
A representation of a group G is a homomorphism from G to a group of operator on a vector space V. The dimension of the representation is the dimension of the vector V.
If D(g) is a matrix...
Hi Guys, :smile:
Can someone please explain to me the logic behind the representation of 'Magnetic Hysteresis loss' as a resistance in electrical equivalent circuits?... will be extremely grateful.
I have studied some info on this subject on the net. Even though the physics of Hysteresis...
Hi Guys, :smile:
Can someone please explain to me the logic behind the representation of 'Magnetic Hysteresis loss' as a resistance in electrical equivalent circuits?... will be extremely grateful.
Thanks & Regards,
Shahvir
Hello Everyone.I want to know if there is anything special about the position or the momentum representation in quantum mechanics.Every book deals with them.Why do not they work with Energy representation or time representation?Do not they exist?Basically,I feel it hard to imagine that a...
Hello,
I hope it's not the wrong forum for my question which is the following:
Is there some list of Lie algebras, whose adjoint representations have the same dimension as their basic representation (like, e.g., this is the case for so(3))? How can one find such Lie algebras? Could you...
Draw the phasor representation for each of the following signals. Also write the signal as one sinusoid, v(t) = 20cos(200t+45°)+cos(200t)
From this equation, do I have to turn to v(t) = A sin (ωt + φ) ? If yes, how can I convert it?
Thank you.
Dear All,
I'm reading Georgi's text about Lie algebra, 2nd edition.
In chap 6, he introduced "Roots and Weights". What I didn't understand is the discussion of section 6.2 about the adjoint representation. He said: "The adjoint representation, is particularly important. Because the rows...
Homework Statement
Let T: P2 > P3 denote the function defined by multiplication by x :T(p(x)) = xp(x). In other words, T(a+bx+cx2) = ax+bx2+cx3
(a) Show that T is a linear transformation.
(b) Find the matrix of T with respect to the standard bases {1,x,x2} for P2 and {1,x,x2,x3} for P3...
Guys,
Quick one, When you do phasors for AC voltages and current... is it theVrms and Irms or can it be Vmax and Imax..? I mean the magnitude of them...I am bit confused
To any teachers or students, either instructing or taking, a Calculus-based Physics I course:
I tutor a calculus-based general physics course in kinematics, and similar topics, and, I recently had a student approach me about his inability to grasp the scalar/dot product, in vector operations...
Homework Statement
i) Show that the Lorentz group has representations on any space \mathbb{R}^d
for
any d = 4n with n = 0, 1, 2, . . .. Show that those with n > 1 are not
irreducible. (Hint: here it might be useful to work with tensors in index
notation and to think of symmetry...
Say I have a matrix similar to the SO(3) matrix for general 3-D rotations, except it has slightly different (simpler) elements, and the symmetry is as follows:
\left(\begin{array}{ccc}
A & B & C \\
B & D & E \\
C & E & D
\end{array}\right) ,
with A, B, C, D, and E all involving somewhat...
I had a question about the following theorem.
Basis Representation Theorem: Let k be any integer larger than 1. Then, for each positive integer n , there exists a representation
n = a_{0}k^{s} + a_{1}k^{s-1} + ... + a_{s}
where a_{0} \neq 0 , and where each a_{i} is...
In quantum mechanics, a free particle is described by a continuous superposition of wavefunctions, which can be done equivalently in real or momentum space. We can look at a particle's probability distribution in real space, take its Fourier transform, and obtain the particle's distribution in...
hey, this is my first time posting, my question is find the power series representation for x/(1-x)^2
I know the representation for 1/1-x is x^n so does that mean x/(1-x)^2 is x^n^2? could use some clarification please
Homework Statement
a) Write a Matlab function, which accepts the following inputs
-a finite set of Fourier series coefficients
-the fundamental period T of the signal to be reconstructed
-a vector t representing the times for which the signal will be reconstructed
Homework...
Hi Guys,
I was wondering if it is possible (why or why not) to define the floor function, Floor[x], as an infinite Taylor Series centered around x=a?
Any sort of help is greatly appreciated!
flouran
Homework Statement
This has been driving me insane, and I'm sure it's something mind-boggling obvious but I can't seem to find it. I'll go through the work through here, I'm trying to prove that
\int_0^1{x^xdx}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^n}.
2. The attempt at a solution...
For carbohydrates, the \alpha position is to the right in a Fischer projection and down in a planar representation. Is it axial or equatorial in a chair conformation?
My book also says that all the OH groups in \beta-D-glucose are in axial positions. Is this true?
Hi, I'm very new on Group Theory, and lacking of easy to understand document on it.
I can't get Representation of SO(3) Groups.
Is there anyone can tell me useful information about it?
Thanks,
Tore Han
Homework Statement
given that X(operator) and P (operator) operate on functions,and the relation [X,P]=ih/2π,show that if X(operator)=x ,and P (operator) has the representation P=-ih/2π*∂/∂x +f(x)
where f(x) is an arbitrary function of x
Homework Equationsquantum mechanic by Liboff...
can someone explain to me that is a spinor and how do I calculate the spin operators from it?
for ex. (from homeword)
the spinor is (|a|*e^(i*alpha), |b|*e^(i*beta))
URGENT: QM Series Representation of Bras, Dirac Brackets
Homework Statement
Suppose the kets |n> form a complete orthonormal set. Let |s> and |s'> be two arbitrary kets, with representation
|s> = \sum c_n|n>
|s'> = \sum c'_n|n>
Let A be the operator
A = |s'><s|
a) Give the...
Homework Statement
Prove:
Representation of vectors by any basis is unique.
Homework Equations
The Attempt at a Solution
The minimal span set and the maximum linearly independent set gives a basis.
PLEASE HELP ! THANK YOU (:
1. My teacher gave us a drawing with 4 vectors on it. Vector A = 130 N and is at a 20 degree angle. Vector B = 100 N and is at a 70 degree angle. Vector C = 70 N and is on the x-axis. Vector D = 50 N and is at a 10 degree angle. Given this picture we are told to...
Homework Statement
Let G be the group S_3. Find the permutation representation of S_3. (Note: this gives an isomorphism of S_3 into S_6)
The Attempt at a Solution
Is there only ONE permutation representation, because the question asks for "the" p.r.
I don't know where to start.
Hi there!
As most people already might know, we can decompose the 27 dimensional representation for the baryons under SU(3) flavour symmetry as 27 = 10 + 8 + 8 + 1. I can find a lot of information about the particles that lie in the decuplet and in the octet, but nothing about which particle...
I'm taking a course in control theory, and have been wondering for a while what the benefits are when you describe a system based on the Laplace method with transfer functions, compared to when you use the state space representation method. In particular, when using the Laplace method you are...
The Wikipedia article on Real Numbers says every "a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way" I don't think this is right. Doesn't there have to be some real numbers that cannot be named, even if we allow that...