On how to diagonalize a Casimir element

Question

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}$I'm trying to read the physics paper Two Dimensional QCD as a String Theory. 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.

Answer 1

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. 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}

Answer 2

First of all, you're right that by "the Casimirs" the authors mean the eigenvalues of the quadratic Casimir operator on the irreps in question — this is a common phrasing in the physics literature.

For $\mathfrak{su}(n)$, the Young diagram with $m$ rows of lengths $n_i$ corresponds to the highest weight $\mu=\sum_i n_i\lambda_i$ (cf. e.g. these lecture notes). From there, the Weyl dimension formula cited in your comment should give the dimensions. For the Casimir eigenvalues, these notes may be of help. Note that there may be factors of 2 (and possibly of the dimension of the irrep) by which conventions used in the physics and mathematics literature may differ in defining the eigenvalues.