$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}$I'm trying to read the physics paper [Two Dimensional QCD as a String Theory](https://arxiv.org/abs/hep-th/9212149). I'm struggling with my ignorance about some computational aspects regarding Lie algebras.


Section 2.3 of the aforementioned paper states:



> 
> The representations of $\U(N)$ are labeled by the
>  Young diagrams, with
>  $m$ ($m\leq N$) boxes of length $n\_{1}\geq n\_{2} \geq \dotsb n\_{m} \geq 0$. Such a representation $R$ has dimensions $d\_{R}$ and Casimirs $C\_{2}^{\U(N)}(R)$ given by \begin{gather\*}
> C\_{2}^{\U(N)}(R)=N\sum\_{i=1}^{m}n\_{i} + \sum\_{i=1}^{m}n\_{i}(n\_{i}+1-2i); \\
> d\_{R}=\frac{\Delta(h)}{\Delta(h^{0})}, \\
> \Delta(h)=\prod\_{1\leq j \leq N}(h\_{i}-h\_{j}), h\_{i}=N+n\_{i}-1, h\_{i}^{0}=N-i.
> \end{gather\*}
> 
> 
> 


**Clarification:** The statement *"$C\_{2}^{\U(N)}(R)$ are Casimirs of the group"* does not make sense because of the fact that the quadratic Casimir element $C\_{2}$ of $\U(N)$ is by definition a bilinear form on the universal enveloping algebra of the Lie algebra of $\U(N)$ and the $C\_{2}$ shown in the paper is a number. I suppose that what the author write as $C\_{2}^{\U(N)}(R)$ is the eigenvalue of $C\_{2}^{\U(N)}$ associated to the representation $R$ labeled by the partition $(n\_{1},\dotsc,n\_{m})$.


**The question:**


I'm asking for your kind help to identify some references where I can learn how to derive the formulas from above and possibly for other cases such as $\SU(N)$, $\SO(N)$ or some symplectic groups if possible.


I'd even be happy if someone could recommend a physical derivation of the formulas.



On how to diagonalize a Casimir element
See <https://arxiv.org/abs/0807.3696> for derivation.


Let $E\_i^j$ be the ${\rm U}(N)$ generators and $V$ be the fundamental representation. Decompose $V^{\otimes k}$ by the Schur-Weyl duality and apply the second Casimir $C\_2 = E\_i^j E\_j^i$ to $V\_R^{{\rm U}(N)} \otimes V\_R^{S\_k}$.


After some computation, $C\_2$ becomes a sum over the permutations $\sum\_{r \neq s} (rs)$. This is a sum of Jucys-Murphy elements whose eigenvalues are the contents of the box; see [Wikipedia](https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element). By summing the contents over $R$, you can derive the Casimir eigenvalue $C\_2^{{\rm U}(N)}(R)$.


That paper of Gross also gives $C\_2^{{\rm SU}(N)}(R)$. His result agrees with Freudenthal's formula of $C\_2=\langle \mu,\mu+2\rho\rangle$ by recalling that the Dynkin label $(m\_1 , m\_2 , \dots , m\_{N-1})$ is related to the partition $(n\_1, n\_2 , \dots, n\_N)$ by
\begin{equation}
m\_i = n\_{N-i} - n\_{N+1-i} \ge 0.
\end{equation}


