Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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)hook 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