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nLab > Latest Changes: hook-content formula

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- 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
- 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
- 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
- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(n)</annotation></semantics></math>$ , the dimension of the rep corresponding to YT $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>$ is
n!∏u:λhλ(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)
- 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:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo>|</mo><msub><mi>ssYTableaux</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>|</mo></mrow><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><msub><mi>s</mi> <mi>λ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mspace width="-0.16667em"/><mo>=</mo><mspace width="-0.16667em"/><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>N</mi></msub><mspace width="-0.16667em"/><mo>=</mo><mspace width="-0.16667em"/><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></munder><mfrac><mrow><mi>N</mi><mo>+</mo><mi>content</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><mi>ℓ</mi><msub><mi>hook</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\vert ssYTableaux_\lambda(N)\right\vert \;=\; s_{\lambda} \big( x_1 \!=\! 1, \cdots, x_N \!=\! 1 \big) \;\; = \;\; \underset{ (i,j) }{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) } \,. </annotation></semantics></math>$
I am indebted to Abdelmalek Abdesselam for hints. (The combinatorics literature is a bit weird about this.)

diff, v9, current
- CommentRowNumber6.
- CommentAuthornLab edit announcer
- CommentTimeDec 25th 2024
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexFixed formula for hook length and added explanation as to why parenthesis are as such in the formula by detailing the arm and leg length. Leonardo Lovera <a href="https://ncatlab.org/nlab/revision/diff/hook-content+formula/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/hook-content+formula/15">v15</a>, <a href="https://ncatlab.org/nlab/show/hook-content+formula">current</a>

Fixed formula for hook length and added explanation as to why parenthesis are as such in the formula by detailing the arm and leg length.

Leonardo Lovera

diff, v15, current

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nLab > Latest Changes: hook-content formula