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nLab > Latest Changes: hook-content formula

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2021

    starting something

    v1, current

    • 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.

    diff, v2, current

    • 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,

    diff, v3, current

    • 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)Sym(n), the dimension of the rep corresponding to YT λ\lambda is

    n!∏ u:λh λ(u). \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:

    |ssYTableaux λ(N)|=s λ(x 1=1,⋯,x N=1)=∏(i,j)N+content(i,j)ℓhook λ(i,j). \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

  6. 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

1 to 6 of 6

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