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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"?

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  • $\begingroup$ What are $\lambda$ and $\lambda_i$? $\endgroup$
    – Wojowu
    Commented Apr 4, 2022 at 14:06
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    $\begingroup$ I assume $\lambda$ is a partition with parts $\lambda_1 \geq \lambda_2 \geq \dots$, while the parts of the conjugate partition $\lambda'$ are $\lambda'_1 \geq \lambda'_2 \geq \dots$. $\endgroup$
    – Michael Joyce
    Commented Apr 4, 2022 at 14:09
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    $\begingroup$ Presumably there is a Type B/C/D hook-content formula that uses these? Although I have not seen that, and it is not discussed in e.g. Sagan's survey: users.math.msu.edu/users/bsagan/Papers/Old/uyt.pdf. $\endgroup$
    – Sam Hopkins
    Commented Apr 4, 2022 at 14:11

2 Answers 2

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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.

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As I guessed in a comment above, there is apparently a symplectic/orthogonal hook-content formula which uses these notions. See "Hook-content Formulae for Symplectic and Orthogonal Tableaux" by Campbell and Stokke https://doi.org/10.4153/CMB-2011-105-7.

Warning: they seem to have withdrawn the arXiv version of their paper (https://arxiv.org/abs/0710.4155) for reasons that are not clear to me.

EDIT: Regarding priority, I see Campbell and Stokke cite El Samra and King for the formula for the evaluation of the symplectic/orthogonal characters at $(1,1,\ldots,1)$ (i.e., pure counting of tableaux); what they do that is new is the principal evaluation $(1,q,\ldots,q^{n-1})$ (i.e., $q$-counting). Perhaps the arXiv withdrawal is about what Per said: copyright.

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  • $\begingroup$ Sam: I was aware of this. But, what is the motivation? $\endgroup$
    – T. Amdeberhan
    Commented Apr 4, 2022 at 14:38
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    $\begingroup$ I don't understand. The formula itself (counting/$q$-counting a certain kind of tableau) is a very natural analog of the hook-content formula. If that's not a good enough motivation, then I'm not sure how you could see the usual definition of content $j-i$ as motivated... $\endgroup$
    – Sam Hopkins
    Commented Apr 4, 2022 at 14:42
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    $\begingroup$ @T.Amdeberhan A natural reason would be that the referees gave a lot of insightful comments, and the paper went through a big overhaul. But then, the journal might not allow this improved version to be put on arxiv, due to copyright claims. $\endgroup$
    – Per Alexandersson
    Commented Apr 5, 2022 at 6:10
  • $\begingroup$ @SamHopkins: I see your point. I, too, don't see $j-i$ as motivated either. $\endgroup$
    – T. Amdeberhan
    Commented Apr 5, 2022 at 13:29

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