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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 9th 2021

starting something

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 9th 2021

I have added the statement, with all its ingredients.

Not fully proof-read yet, but need to urgently do something else for the moment.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 9th 2021

adding the hyphen to the title, since apparently the name does not refer to “content of hooks” but to “hooks and content” of boxes. I have added a line on this point to the entry,

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMay 12th 2021
• (edited May 12th 2021)

We should have something on the relationship between hook lengths and rep dimensions. So for $Sym(n)$, the dimension of the rep corresponding to YT $\lambda$ is

$\frac{n!}{\prod_{u: \lambda} h_{\lambda(u)}}.$

From a nice efficient set of notes here:

• Yufei Zhao, Young Tableaux and the Representationsof the Symmetric Group, (pdf)
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 16th 2021
• (edited May 16th 2021)

I have now added statement of and references for the actual/standard form of the hook-content formula:

$\left\vert ssYTableaux_\lambda(N)\right\vert \;=\; s_{\lambda} \big( x_1 \!=\! 1, \cdots, x_N \!=\! 1 \big) \;\; = \;\; \underset{ (i,j) }{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) } \,.$

I am indebted to Abdelmalek Abdesselam for hints. (The combinatorics literature is a bit weird about this.)

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