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 ...

  Decomposing a representation under a subgroup

+ 4 like - 0 dislike
1010 views

I am trying to remind myself how decomposing representations works by looking at an easy example.

Using the notation in Slansky, consider the representation of $SU(5)$ that has highest weight $(0\ 1\ 0\ 0)$. How does this rep decompose under $$SU(5)\rightarrow SU(3)\times SU(2)\times U(1)\ ?$$

Could someone please remind me how this works, and maybe mention the general steps along the way as well?


This post imported from StackExchange Physics at 2015-01-16 22:02 (UTC), posted by SE-user Heterotic

asked Jan 16, 2015 in Mathematics by Heterotic (525 points) [ revision history ]
edited Jan 16, 2015 by Dilaton
Related: physics.stackexchange.com/q/159597/2451

This post imported from StackExchange Physics at 2015-01-16 22:02 (UTC), posted by SE-user Qmechanic

After some digging, I found these notes useful.

1 Answer

+ 2 like - 0 dislike

Decomposing representations is the process of breaking down a representation of a group into representations of subgroups. In the case of SU(5) decomposing to SU(3) x SU(2) x U(1), the representation with highest weight (0 1 0 0) will be decomposed into irreducible representations of the subgroups.

Here are the general steps to decompose a representation of a group G into irreducible representations of subgroups H:

  1. Determine the weights of the representation of G you are interested in.

  2. Determine the weights of the representations of H.

  3. Find the highest weight of each irreducible representation of H that appears in the decomposition.

  4. Determine the multiplicities of the irreducible representations of H by counting the number of times each highest weight appears in the decomposition.

  5. Write down the decomposition of the representation of G into a direct sum of irreducible representations of H, including the multiplicities.

It is important to note that in general, the decomposition of a representation of a group into irreducible representations of subgroups is not unique, and the decomposition you obtain may depend on the method you use to perform the decomposition.

answered Feb 6, 2023 by anonymous [ 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$\hbar$ysicsOver$\varnothing$low
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
...