Representations Definition and 201 Threads

Hi, All: I need some help with some "technology" on differential forms, please: 1)Im trying to understand how the hyperplane field Tx\Sigma< TpM on M=\Sigma x S1 , where \Sigma is a surface, is defined as the kernel of the form dθ (the top form on S1). I know that...

Hello, I was just curious if anyone knew of a single place with a list of many different gamma matrix representations, I haven't been able to find what I want by just searching google. In particular, I'm looking for a representation of the 5+1 dimensional Clifford algera. In other dimensions...

Homework Statement Show that B = {x2 −1,2x2 +x−3,3x2 +x} is a basis for P2(R). Show that the differentiation map D : P2(R) → P2(R) is a linear transformation. Finally, find the following matrix representations of D: DSt←St, DSt←B and DB←B. Homework Equations The Attempt at a...

I guess this is a simple question. Say I am tasked with finding the Taylor series for a given function. Well say that the function is analytic and so we know there's a taylor series representation for it. Am I gauranteed that this representation is the only power series representation for it...

Can anyone recommend some litterature on representations of the Lorentz group. I'm reading about the dirac equation and there the spinor representation is used, but I would very much like to get a deeper understanding on what is going on.

I have to choose a total of 12 modules for my 3rd year. I've everything decided except four of them. I want to eventually do research either General Relativity, quantum mechanics, string theory, something like that. I'm torn between Group Representations, with one of Practical numerical...

I'm a college Sophomore majoring in math and over the summer I've been playing around with improper integrals, specifically integrals with limits at infinity because they've always fascinated me. The highest calculus course I've taken is Calc II, so I might be missing something here. Anyways...

"In a Lorentz invariant theory in d dimensions a state forms an irreducible representation under the subgroups of SO(1,d-1) that leaves its momentum invariant." I want to understand that statement. I don't see how I should interpret a state as representation of a group. I have learned that...

I have been reading about the legendre polynomials and how their completeness allows you to write any function as a sum of them. I have seen that used in electrostatics for the multipole expansion, which I guess is pretty nice, but here's the deal: It seems that I am to learn more and more...

I am writing an undergraduate "thesis" on group representations (no original work, basically a glorified research paper). I was wondering if anyone could suggest interesting aspects that might be worth writing about in my paper. I have only just begun to explore the topic, and I see that it...

There seem to be two definitions for a regular representation of a group, with respect to a field k. In particular, one definition is that the regular representation is just left multiplication on the group algebra kG, while the other is defined on the set of all functions f: G \to k . I do not...

Is it true that SO(3) has no complex 2-dimensional representation (except the trivial one...)? How to see this? If it is nontrivial, can someone provide a source? Is there such a thing as a classification of all the linear representations of SO(n)? Thanks

Hi All, I am trying to work through a QFT problem for independent study and I can't quite get my head around it. It is 5.16 from Tom Bank's book (http://www.nucleares.unam.mx/~Alberto/apuntes/banks.pdf) which goes as follows: "Show that charge conjugation symmetry implies that the...

Direct product of two irreducible representations of a finite group can be decomposed into a direct sum of irreducible representations. So, starting from a single faithful irreducible representation, is it possible generate every other irreducible representation by successively taking direct...

I am reading Dummit and Foote Ch 18, trying to understand the basics of Representation Theory. I need help with clarifying Example 3 on page 844 in the particular case of S_3 . (see the attahment and see page 844 - example 3) Giving the case for S_3 in the example we have the...

I am reading James and Liebeck's book on Representations and Characters of Groups. Exercise 1 of Chapter 3 reads as follows: Let G be the cyclic group of order m, say G = < a : a^m = 1 >. Suppose that A \in GL(n \mathbb{C} ) , and define \rho : G \rightarrow GL(n \mathbb{C} ) by...

Hi guys! I still have problem clearing once and for all my doubt on the spinor representation. Sorry, but i just cannot catch it. 1) ----- Take a left handed spinor, \chi_L. Now, i know it transforms according to the Lorentz group, but why do i have to take the \Lambda_L matrices belonging...

I'm having some trouble with a concept in group theory. I'm reading Howard Georgi's book on Lie Algebra, this is from the 1st chapter. Really sorry to have to use a picture but I don't know how to TeX a table: There's a couple things I don't quite understand but mainly, I don't see how he...

There is a Theorem that says FG-Modules are equivalent to group representations: "(1) If \rho is a representation of G over F and V = F^{n}, then V becomes an FG-Module if we define multiplication vg by: vg = v(g\rho), for all v in V, g in G. (2) If V is an FG-Module and B a basis of V...

I just started my first graduate level controls class this semester, and it looks like my professors notes aren't going to be quite enough. Can anybody recommend a good book with lots of state space representation examples, or if not a book, a website would do fine as well. Thanks for any help.

For spin 1/2 particles, I know how to write the representations of the symmetry operators for instance T=i\sigma^{y}K (time reversal operator) C_{3}=exp(i(\pi/3)\sigma^{z}) (three fold rotation symmetry) etc. My question is how do we generalize this to, let's say, a basis of four...

I am a physicist, so my apologies if haven't framed the question in the proper mathematical sense. Matrices are used as group representations. Matrices act on vectors. So in physics we use matrices to transform vectors and also to denote the symmetries of the vector space. v_i = Sum M_ij...

I happen to be studying the basics of quantum mechanics at the moment and have made acquaintance with the vector representation of quantum states, in particular the two states of electron spin. For this question let's just say the spin can be up or down. The state of the spin is...

Hi According to the Kraus representation theorem, a map \mathcal{E}: \text{End}(\mathcal{H}_A) \rightarrow \text{End}(\mathcal{H}_B) is a trace-preserving completely positive map if and only if it can be written in an operator sum representation \mathcal{E}: \rho \mapsto \sum_k A_k \rho...

Heres the reducible representation made by counting the number of bonds left unchanged by each symmetry operation of water: http://img808.imageshack.us/img808/704/red0.png and here's the irreducible representations extracted from it: http://imageshack.us/m/695/3829/red01l.png in the book...

This is something I feel I should know by now, but I've always been very confused about. Specifically, how does one determine what each representation of the Lorentz group corresponds to? I mean, I know that the (1/2,0) and the (0,1/2) representations correspond to right and left handed spinors...

Hi everyone! I really need your help if you are good in time series. I have a problem on moving average representations. I attach the problem description; also, I attach my attempt to solve it. Cannot go any further. Please please help me. Thank you!

Typically I understand that projection operators are defined as P_-=\frac{1}{2}(1-\gamma^5) P_+=\frac{1}{2}(1+\gamma^5) where typically also the fifth gamma matrices are defined as \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 and.. as we choose different representations the projection...

Hello all, I'm stuck on understanding part of a discussion of representations and Clebsch-Gordan series in the book 'Groups, representations and Physics' by H F Jones. I'd be grateful to anyone who can help me out. For starters, this discussion is in the SU(2) case. I don't know how to...

I don't know how to do the following homework: Let Gbe a finite group and let \rho : G \rightarrow GL(E)be a finite-dimensional faithful complex representation, i.e. ker \rho = 1. For any irreducible complex representation \piof G, show that there exists k \geq 1 such that \pi is an...

Hi, I'm a newbie here, i joined just now purely to ask this question that's been on my mind recently. Now i apologise if this question is fundamentally wrong (which it probably is), but I'm only the average person with an amateur interest in physics :P So don't laugh. Firstly, as you know, we...

I'm doing a course which assumes knowledge of Group Theory - unfortunately I don't have very much. Can someone please explain this statement to me (particularly the bits in bold): "there is only one non-trivial irreducible representation of the Cliford algebra, up to conjugacy" FYI The...

I have a very superficial understanding of this subject so apologies in advance for what's probably a stupid question. Can someone please explain to me why if we have a Lie Group, G with elements g, the adjoint representation of something, eg g^{-1} A_\mu g takes values in the Lie Algebra of G...

I don't understand how to find the irreducible representations of a group. Under transformation U: (T')^{ijk}=U^{il}U^{jm}U^{kn}T^{lmn} But suppose (M')^{ijk}=(T')^{ijk}+\pi (T')^{jik} Then (M')^{ijk}=U^{il}U^{jm}U^{kn}T^{lmn} +\pi U^{jl}U^{im}U^{kn}T^{lmn}=U^{il}U^{jm}U^{kn}(T^{lmn}+\pi...

What I am trying to do is start with a Dynkin diagram for a semi-simple Lie algebra, and construct the generators of the algebra in matrix form. To do this with su(3) I found the root vectors and wrote out the commutation relations in the Cartan-Weyl basis. This gave me the structure constants...

I'm a mathematician, and I have trouble understanding the physics notation. I'm glad if someone could help me out. Here's my question: Let g be a Lie algebra and r_1: g -> End(V_1), r_2: g -> End(V_2) be two representations. Then there is a representation r3:=r_1 \otimes r_2: g -> End...

I'm studying the representations of SU(2,1) [or U(1,1)], and since they are non-compact, their representations are necessarily infinite dimensional. I have a couple questions. In the literature, they say the algebra satisfied by the three generators, T_1, T_2, T_3 is [T_1,\,T_2]=-i T_3...

Hi all, It's a well-known result that any finite or compact group G admits a finite-dimensional, unitary representation. A standard proof of this claim involves defining a new inner product of two vectors by averaging the inner products of the images of the vectors under each element of the...

Hi, I had a question about irreducible representations and elementary particles... Namely, I've been told by teachers and read in a few texts that particles ARE irreducible representations, and I have never been able to wrap my mind around what that means. Please keep in mind that I am no...

Show that representations of the angular momentum algebra [J_i, J_j ] = \epsilon_{ijk}J_k act on finite-dimensional vector spaces, V_j , of dimension 2j + 1, where j = 0, 1/2, 1, \dots This sounds incredibly easy but what is the question actually asking me to do?

Say you have the coefficients a_k of a Fourier series representation of some function x(t). You can easily then give x(t) as $$x(t) = \sum_{k = -\infty}^{\infty} a_k e^{i k \omega_0 t}$$ But this doesn't do much good in telling you what the actual function looks like. For example, if we have...

So I'm studying molecules and symmetry and I was wondering if there was a intuitive way of understanding why there are as many irreducible representations as there are classes. I keep getting lost in the math of the characters.

Hi, I'm currently reading "Supersymmetry demystifed" by Patrick Labelle, chapter 10, about SUSY non-Abelian gauge theories. We have a Lagrangian with SU(N)-gauge fields, and gaugino's. What puzzles me are the following claims of Labelle about the representations. In the...

Hi, this isn't a homework question per se (it's the summer hols, I'm between semesters) but it's something that I never really got during the QM module I just did. I found myself blindly calculating exam & homework problems, and just feel like this is some stuff I should get cleared up...

how to construct an irreducible representation of SU(3)?

Hi, hopefully this is the right board to ask this on. I'm currently reading Groups, representations and Physics by Jones, and trying to get my head around induced transformations of the wavefunction. The problem is I seem to understand nearly all of what he's saying except the crucial part I...

I'm just having a little trouble getting my head around how representation theory works. Say for example we are working with the dihedral group D8. Then the degrees of irreducible representations over C are 1,1,1,1,2. So there are 4 (non-equivalent) irreduible representations of degree 1...

Hello, in relativistic quantum field theories all particles are members of (unitary) representations of the Poincare group. For massive particles m² > 0 one gets the usual scalar / spinor / vector representations with spin J = 0, 1/2, 1, 3/2, ... and dim. rep. = 2J+1. For massless particles...

I have a very basic query about multiplets. In the SU(3) approach strongly interacting particles, quarks and hadrons are the basis vectors of irreducible representations of SU(3). Now, quarks and hadrons are definite properties with define eigenvalues of hypercharge and isospin: to put it...

Is there any book or source avaliable that clearly shows the point symmetry operation with matrix representations?