Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Decomposition of group representation using tensor method

+ 3 like - 0 dislike
1729 views

I am dealing with the decomposition of the representation $5\otimes5$ of $SU(5)$:

$$5\otimes5=15\oplus10 $$

demonstration:

$$u^iv^j=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}(u^iv^j-u^jv^i)=$$

$$=\frac{1}{2}(u^iv^j+u^jv^i)+\frac{1}{2}\epsilon^{ijxyk}\epsilon_{xyklm}u^lv^m$$

where the term $\frac{1}{2}(u^iv^j+u^jv^i)$ has 15 independent components and the other has 10 components.

My question is: being the $\epsilon^{ijxyk}$ invariant in $SU(5)$, shouldn't the tensor $\epsilon_{xyklm}u^lv^m$ transform under the $\overline{10}$ representation, having 3 low free index?

(according to my notation an upper index transform under the $D$ representation while a lower index transforms under the $\overline{D}$ representation).


This post imported from StackExchange Physics at 2015-03-12 12:20 (UTC), posted by SE-user Caos

asked Mar 11, 2015 in Mathematics by Caos (15 points) [ revision history ]
retagged Mar 12, 2015

I want to attract your attention to the fact that this "decomposition" is kind of illusory. Indeed, if $u$ and $v$ belong to different 5D spaces and the indices $i$ and $j$ are fixed, the "direct" product $u^iv^j$ is simply proportional to each of the fixed elements, say, to $u^3$ and $v^5$ whereas the first decomposed expression "involves" other elements, namely $u^5$ and $v^3$. Thus the second addendum in the decomposition formula serves to subtract extra terms from the first addendum :-)

The true decomposition is only valid for $u^i u^j$.

1 Answer

+ 3 like - 0 dislike

It's a convention which of these reps is called ${\bf 10}$ and which is called $\overline{\bf 10}$. The convention that the people choose is arguably the simpler one among the two: ${\bf 10}$ is the antisymmetric product of two ${\bf 5}$, i.e. ${\bf 5}\wedge{\bf 5}$, which are also without bars.

With that choice, one can prove that $\overline{\bf 10}$ which is defined as the complex conjugate representation is either an analogous representation where the upper indices are replaced with lower ones or vice versa; or it is ${\bf 5}\wedge {\bf 5}\wedge{\bf 5}$.

It is simply a mathematical fact (which you have implicitly proved, using the epsilon symbol) that ${\bf 5}\wedge{\bf 5}$ is the complex conjugate of the representation ${\bf 5}\wedge {\bf 5}\wedge{\bf 5}$ – so if one of them is called without the bar, the other one must be with the bar, and the dominant convention is one written above.

This post imported from StackExchange Physics at 2015-03-12 12:20 (UTC), posted by SE-user Luboš Motl
answered Mar 11, 2015 by Luboš Motl (10,278 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\varnothing$ysicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...