Recent content by rkrsnan

Ok, I was wrong, I tried with random unitary matrices and it didn' t work. So let me modify my hypothesis. Let U be a discrete unitary matrix, with U^n = I, where n is an integer. Its eigenvalues will be of the form e^{i m\frac {2\pi} {n}} where m is an integer. There exists a basis and...

Ohh, I now notice something. Since my representation matrices are unitary, their eigenvalues are always in the form e^{i \theta}. For the matrix that I wrote in the first post, the eigenvalue (corresponding to the eigenvector (1,\omega, \omega^2)) is e^{i 2\pi/3}. So basically the "appealing"...

Thanks guys for the reply. I am aware of the things you pointed out. As Erland said I should have been more specific on what I want. I am studying the representations of discrete groups for application in physics. Eigenvectors of group elements are significant in this context (for example the...

I am wondering if there is a systematic way to fix the phase of complex eigenvectors. For example e^{i \theta}(1,\omega,\omega^2) where e^{i \theta} is an arbitrary phase and \omega and \omega^2 are the cube roots of unity, is an eigenvector of the cyclic matrix \left(\begin{matrix}0&...

Yep, understood. About the cyclic center and having faithful irreducible reps, does this result work if the center is identity? I can find examples of groups in which center is identity; in some cases faithful irreps exist and in some others it doesn't.

Thank you so much! That was totally what I wanted to know. PS: Yes, I should have written 'tensor product' instead of 'direct product'.

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...

Yes, I am only concerned with irreducible representations. I didn't know about the subtlety of the degenerate/nondegenerate bilinear forms though. Thanks very much for the reply and the help.

My apologies for the confusing terminology. What I meant was a complex representation which has real characters. It is also called a quaternionic representation. I also understand that for a pseudo real representation the Frobenius–Schur indicator will be -1. From wikipedia: "It is −1 exactly...

Can you please give an example for a finite group with a three dimensional pseudo real representation? I can find examples of finite groups with 2, 6 and 8 dimensional pseudo real representations, but couldn't find any with a three dimensional pseudo real rep. Is there some theorem that states...

For matrices, Schur product or Hadamard product is defined as the entry wise product. I want to know if they have a similar type of multiplication for complex numbers. That is (a+ i b) o (c + i d) = (a c + i b d) I encounter a situation where such a definition is useful. In physics I get...

Using a general 4 parameter 2x2 unitary matrix \begin{pmatrix} |v_{1}> \\ |v_{2}> \end{pmatrix} =\begin{pmatrix} e^{\phi 1} e^{\phi 2} & 0 \\ 0 & e^{\phi 1}\end{pmatrix} \begin{pmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta\end{pmatrix}\begin{pmatrix} e^{\phi 3} & 0 \\ 0 &...

What do you mean by constraint? Plz elaborate Wikipedia article is reasonably good.

Not only the center of the detector, but other regions like the beam pipe are kept in vacuum. The detector set up usually consists of magnetic fields. It is used to bend the collision products and measure their momenta. The effect of this field on the interaction between particles during...

Charged leptons do not undergo oscillations simply because their definition does not allow oscillations. In standard model, charged leptons are defined as the eigenstates of weak interaction having definite mass. A neutrino is also a weak eigenstate, but it is superposition of states having...