# Representation theory of $SO(n)$

This is probably not a very ethical question to ask but I need to have a fast introduction to a range of concepts about the representation theory of the $SO(n)$ and I would be happy to see some online references which will help me do this journey.

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I am listing below the specific concepts about it that I need to know the most.

• I want to know about the concept of "lowest weights" and "highest weights" and how a string of $[ \frac{n}{2}]$ numbers (integers?) say $(h_i, i = 1,2,..,[ \frac{n}{2}])$ label a representation of $SO(n)$.

• I want to know what is a "quadratic Casimir" of such a representation (vaguely I understand that to be the eigenvalue of an operator which commutes with all the basis of the group's Lie algebra). For such a representation as above the quadratic Casimir is $$c_2(\{h_i\}) = \sum _{i=1} ^{ i= [ \frac{n}{2}]} \left(h_i ^2 + (n-2i)h_i\right)$$

which will help see that the quadratic Casimir for a "scalar representation" is $0$, for a "vector representation" it is $n-1$ and for a "spinor representation" it is $\frac{n(n-1)}{8}$

I hope to know how to convert the above in quotes standard physics terminology into the language of weights.

• If $\{ H_i\}$ form a set of Cartan generators of the group $SO(2n+1)$ then in the "vector representation" the character of the element labelled by the real numbers say $\{t_i\}$ is $1+\sum _{i=1} ^n 2\cosh(t_i)$ and for the "spinor representation" it is given by $1+\prod _{i=1} ^n 2\cosh(t_i)$ and in general it is given as,

$$\chi (h_i,t_i) = \frac{\det \left(\sinh [ t_i(h_j +(n-j) +\frac{1}{2} ]\right) }{\det \left(\sinh [t_i((n-j) +\frac{1}{2}\right) }$$

• The above equation leads to a Clebsch-Gordan (which I am familiar for $SO(3)$) like thinking that $\{h_i\} \times \{\mbox{vector}\} = \{h_i\} +$ All representations obtained from $\{h_i\}$ by adding or subtracting unity from a single $h_i$ such that in the resulting set $h_n \geq 0$ and $h_i \geq h_{i+1}$ for other $is$. And similarly for $\{h_i\} \times \{\mbox{spinor}\} = \{h_i\} +$ All representations obtained from $\{h_i\}$ by adding or subtracting half from every $h_i$ such that in the resulting set $h_n \geq 0$ and $h_i \geq h_{i+1}$ for other $is$

• Similarly for $SO(2n)$ the corresponding character formula looks like,

$$\chi (h_i,t_i) = \frac{\det \left(\sinh [ t_i(h_j + n-j)]\right) + \det \left(\cosh [ t_i(h_j + n-j)]\right) }{\det \left(\sinh [t_i(n-j)] \right) }$$

And similar interpretations lead to the thinking that $\{h_i\} \times \{\mbox{vector}\} = \{h_i\} +$ All representations obtained from $\{h_i\}$ by adding or subtracting unity from a single $h_i$ such that in the resulting set $\vert h_n \vert \geq 0$ and $h_i \geq h_{i+1}$ for other $is$. And similarly for $\{h_i\} \times \{\pm \mbox{chirality spinor}\} = \{h_i\} +$ All representations obtained from $\{h_i\}$ by adding or subtracting half from every $h_i$ with number of subtractions being even/odd, such that in the resulting set $\vert h_n \vert \geq 0$ and $h_i \geq h_{i+1}$ for other $is$

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I am not sure I am reading everything right but I hope to get corrected and I would be happy to see references hopefully online which will explain to me the above things.

## 1 Answer

Read the book by Fulton and Harris on Representation Theory.

• A little ingenuity online should be enough to get hold of an electronic copy... On the other hand, I am of the opinion that "fast" is a requirement rather hard to satisfy! – Mariano Suárez-Álvarez Feb 5 '11 at 21:13
• Well, every minimally good library should have that book. – Mariano Suárez-Álvarez Feb 6 '11 at 15:26
• Regarding the downvote: I am really sorry if this answer sounds too harsh, but math.SE is not the correct place to ask this kind of questions which amounts to «please explain the represnetation theory of SO(n) to me» and to which not even a whole seminar would provide a complete answer. The book by Fulton and Harris is a 500-page answer to this question, and it is an amazingly good answer at that. For such a broad question I honestly think that any attempt at providing an answer in these site is a waste of effort. – Mariano Suárez-Álvarez Feb 17 '12 at 20:37
• Sure you are right. At the time when I asked this question I didn't know as to how deep the theory went to answer this question and I was given the impression by certain people that there is a quick answer to this. At that time I needed to urgently understand this. Over time since then - almost an year ago - I have made some progress with this theory - studying it off and on - but I still think that it would be great if there does exist any resource which gives a bare-handed quick calculational guide through this. – Anirbit Feb 18 '12 at 22:56
• The book by Fulton and Harris is exactly that! And, while it is not short, it is extraordinarily friendly and easy to read. – Mariano Suárez-Álvarez Feb 18 '12 at 23:55