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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something <a href="https://ncatlab.org/nlab/revision/hook+content+formula/1">v1</a>, <a href="https://ncatlab.org/nlab/show/hook+content+formula">current</a>

   starting something

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the statement, with all its ingredients. Not fully proof-read yet, but need to urgently do something else for the moment. <a href="https://ncatlab.org/nlab/revision/diff/hook+content+formula/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/hook+content+formula/2">v2</a>, <a href="https://ncatlab.org/nlab/show/hook+content+formula">current</a>

   I have added the statement, with all its ingredients.

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

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 9th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadding 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, <a href="https://ncatlab.org/nlab/revision/diff/hook-content+formula/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/hook-content+formula/3">v3</a>, <a href="https://ncatlab.org/nlab/show/hook-content+formula">current</a>

   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,

   diff, v3, current

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 12th 2021
   • (edited May 12th 2021)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWe 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](https://yufeizhao.com/research/youngtab-hcmr.pdf))

   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)
5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 16th 2021
   • (edited May 16th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.) <a href="https://ncatlab.org/nlab/revision/diff/hook-content+formula/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/hook-content+formula/9">v9</a>, <a href="https://ncatlab.org/nlab/show/hook-content+formula">current</a>

   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.)

   diff, v9, current