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.
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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 $SO(n)$.
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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,
$$\chi (h_i,t_i) = \frac{\det \left(\sinh [ t_i(h_j +(n-j) +\frac{1}{2} ]\right) }{\det \left(\sinh [t_i((n-j) +\frac{1}{2}\right) } $$
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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 $is$
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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.
Best Answer
Read the book by Fulton and Harris on Representation Theory.