Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda\_1,\lambda\_2,\dots)$ and cells are in the Young diagram.


The **symplectic content** of cell $(i,j)$ of $\lambda$ is defined by
$$c\_{sp}(i,j)=\begin{cases} \lambda\_i+\lambda\_j-i-j+2 \qquad \text{if $i>j$} \\
i+j-\lambda\_i'-\lambda\_j' \qquad \qquad \text{if $i\leq j$}.\end{cases}$$


The **orthogonal content** of cell $(i,j)$ of $\lambda$ is defined by
$$c\_{O}(i,j)=\begin{cases} \lambda\_i+\lambda\_j-i-j \qquad \qquad \text{if $i\geq j$} \\
i+j-\lambda\_i'-\lambda\_j'-2 \qquad \text{if $i< j$}.\end{cases}$$


Although I have used these in my analysis, I still wonder:



> 
> **QUESTION.** What is the motivation behind these definition choices for the "contents"?
> 
> 
> 



What is the motivation behind symplectic/orthogonal content?
A hook-content formula, using the contents $c\_{sp}(i,j)$ and $c\_O(i,j)$, for the dimensions of the irreducible polynomial representations of the symplectic and orthogonal groups, goes back to Ron King. I believe the relevant paper is <https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00053086>, but I did not check for sure.


