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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,847 answers , 20,601 comments
1,470 users with positive rep
501 active unimported users
More ...

Recipe to compute dimension and decompose product of $SO(N)$ group representations

+ 7 like - 0 dislike
232 views

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts:

  1. There is a correspondence between irreps of $SU(N)$ and YT;

  2. There is an easy way to compute dimension of a certain irrep of $SU(N)$ (using again YT);

  3. There is an easy algorithm to multiply two or more irreps of $SU(N)$ in terms of YT;

Questions: Is there a way to do the same (points 1. 2. 3.) with $SO(N)$? What are the algorithms to do that?


This post imported from StackExchange Mathematics at 2015-04-29 18:23 (UTC), posted by SE-user MaPo

asked Apr 28, 2015 in Mathematics by MaPo (60 points) [ revision history ]
edited Apr 29, 2015 by Dilaton

1 Answer

+ 4 like - 0 dislike

1) There is a correspondence, but there are more restrictions on YT than in the case of $SU(N)$ where the restriction is not to have more than N-1 boxes in a column. Due to an additional invariant tensor, metric $\delta^{ab}$, we can take traces and irreducible tensors need to be traceless. Combining it with the $\epsilon$-symbol you get that the total height of the first two columns has to be not greater than $N$, otherwise the tensor is identically zero. 

2) There is a generalization of the Hook formula that works for $SU(N)$. The details can be found in this old paper.

3) Again there is an algorithm, which is a superposition of two $SU(N)$ tensor product rules. One has to take into account traces. The precise formulas are the very first in this paper

answered May 3, 2015 by JohnS (180 points) [ revision history ]

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$\hbar$ysicsOve$\varnothing$flow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...