nLab
hook length formula

Contents

Context

Combinatorics

Representation theory

Contents

Idea

Given a partition/Young diagram λ\lambda with nn boxes/of nn:

the hook length formula expresses both

in terms of the lengths of all “hooks” inside the Young diagram;

similarly, the hook-content formula expresses both

in terms of the length of all hooks and the “content” of all of the boxes.

hook length formulahook-content formula
number of standard Young tableauxnumber of semistandard Young tableaux
dimension of irreps of Sym(n)dimension of irreps of SL(n)

Preliminaries

Given a Young diagram, the hook at any one of its boxes is the collection of boxes to the right and below that box, and including the box itself. We write “hook λ\ell hook_\lambda” for the length of such a hook, i.e. for the number of boxes it contains. Formally:

Definition

(hook length)
Let λ=(λ 1λ rows(λ))\lambda = (\lambda_1 \geq \cdots \geq \lambda_{rows(\lambda)}) be a partition/Young diagram. Then for

  • i{1,,rows(λ)}i \in \{1, \cdots, rows(\lambda)\},

  • j{1,,λ i}j \in \{1, \cdots, \lambda_i\}

the hook length at (i,j)(i,j) is

hook λ(i,j)1+(λ ij)+(λ j1), \ell hook_\lambda(i,j) \;\coloneqq\; 1 + (\lambda_i - j) + (\lambda'_j - 1) \,,

where λ\lambda' denotes the conjugate partition (see there).

Definition

(numbers of (semi-)standard Young tableaux)
Given a partition λPart(n)\lambda \in Part(n), and a positive natural number N +N \in \mathbb{N}_+, consider

  • the number of standard Young tableaux:

    (1)|sYTableaux λ| + \left\vert sYTableaux_\lambda\right\vert \;\in\; \mathbb{N}_+
  • the number of standard Young tableaux with bounded entries T ijNT_{i j} \leq N:

    (2)|ssYTableaux λ(N)| + \left\vert ssYTableaux_\lambda(N)\right\vert \;\in\; \mathbb{N}_+

of shape λ\lambda.

Details

Counting standard Young tableaux

Proposition

(hook length formula for standard Young tableaux)
Given a partition (Young diagram) λ\lambda of nn (boxes), the number (1) of standard Young tableaux of shape λ\lambda equals the factorial of nn over the product of the hook lengths (Def. ) at all the boxes of λ\lambda:

(3)|sYTableaux λ|=n! (i,j)hook λ(i,j). \left\vert sYTableaux_\lambda \right\vert \;=\; \frac{ n! }{ \prod_{(i,j)} \ell hook_\lambda(i,j) }.

This is due to Frame, Robinson & Thrall 54. Textbook accounts include Stanley 99, Cor. 7.21.6, Sagan 01 Thm. 3.10.2.

Measuring dimension of irreps of Sym(n)Sym(n)

The dimension of the irrep of the symmetric group Sym(n)Sym(n) that is labelled by a given Young diagram λ\lambda (the Specht module S (λ)S^{(\lambda)}, see at representation theory of the symmetric group) equals the number of standard Young tableaux of shape λ\lambda

dim(S (λ))=|sYTableaux λ| dim(S^{(\lambda)}) \;=\; \left\vert sYTableaux_\lambda \right\vert

(e.g. Sagan, Thm. 2.6.5)

and hence is also given by the hook length formula (3):

(4)dim(S (λ))=n! (i,j)hook λ(i,j). dim(S^{(\lambda)}) \;=\; \frac{ n! }{ \prod_{(i,j)} \ell hook_\lambda(i,j) } \,.

This is actually the statement of Frame, Robinson, & Thrall 54, Thm. 1. Textbook accounts include James 78, Thm. 20.1.

Counting semi-standard Young tableaux

Proposition

(hook length formula for semi-standard Young tableaux)
Given

the number (2) of semi-standard Young tableaux of shape λ\lambda and entries N\leq N (hence the value of the Schur polynomial s λ(x 1,,x N)s_\lambda(x_1, \cdots, x_N) at x i=1x_i = 1) is:

(5)|ssYTableaux λ(N)|=s λ(1,,1Narguments)=(i,j)Ni+jhook λ(i,j). \left\vert ssYTableaux_\lambda(N) \right\vert \;=\; s_\lambda \big( \underset{ \mathclap{ N\; arguments } }{ \underbrace{ 1, \cdots, 1 } } \big) \;=\; \underset{(i,j)}{\prod} \frac{ N - i + j }{ \ell hook_\lambda(i,j) } \,.

Here

content(i,j)ji content(i,j) \;\coloneqq\; j - i

is also called the content of the box (i,j)(i,j), whence (5) is also called a hook-content formula (“hook length and box content”):

|ssYTableaux λ(N)|=s λ(1,,1Narguments)=(i,j)N+content(i,j)hook λ(i,j). \left\vert ssYTableaux_\lambda(N) \right\vert \;=\; s_\lambda \big( \underset{ \mathclap{ N\; arguments } }{ \underbrace{ 1, \cdots, 1 } } \big) \;=\; \underset{(i,j)}{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) } \,.

Measuring of dimension of irreps of GL(n,)GL(n, \mathbb{C})

The dimension of the irrep V (λ)V^{(\lambda)} of the general linear group GL(n,)GL(n, \mathbb{C}) that is labelled by a given Young diagram λ\lambda (see at representation theory of the general linear group), is also given by the hook-content formula (5):

dim(V (λ))=(i,j)N+content(i,j)hook λ(i,j) dim\big(V^{(\lambda)}\big) \;=\; \underset{(i,j)}{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) }

This appears as Sternberg 94 (C.27)

References

For standard Young tableaux

The original proof is due to:

Textbook accounts:

Further review:

  • Alex Ghorbani, Section 4.4 of: Applications of representation theory to combinatorics (pdf)

  • Yufei Zhao, Section 4.4. of Young Tableaux and the Representations of the Symmetric Group (pdf, pdf)

  • Shiyue and Andrew, Young Tableaux and Probability, 2019 (pdf)

See also:

Alternative proofs:

  • Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, A direct bijective proof of the hook-lengthformula, Discrete Mathematics and Theoretical Computer Science1, 1997, 53–67 (pdf)

  • Kenneth Glass, Chi-Keung Ng, A Simple Proof of the Hook Length Formula, The American Mathematical Monthly Vol. 111, No. 8 (Oct., 2004), pp. 700-704 (jstor:4145043)

  • Jason Bandlow, An elementary proof of the hook formula, The Electronic Journal of Combinatorics 15 (2008) (pdf)

  • Bruce Sagan, Probabilistic proofs of the hook length formulas involving trees, Séminaire Lotharingien de Combinatoire 61A (2009) (pdf)

Generalizations:

  • Ionuţ Ciocan, Fontanine Matjaž, Konvalinka, Igor Pak, The weighted hook length formula, Journal of Combinatorial Theory, Series A Volume 118, Issue 6, August 2011, Pages 1703-1717 (doi:10.1016/j.jcta.2011.02.004)

  • Alejandro Morales, Igor Pak, Greta Panova, Hook formulas for skew shapes I. q-analogues and bijections, Journal of Combinatorial Theory Series A 154 (2018), pp 350–405 (arXiv:1512.08348)

For semistandard Young tableaux

The original proof:

  • Richard Stanley, Theorem 15.3 in: Theory and application of plane partitions 2, Studies in Applied Math. 50 3 (1971), 259-279 (pdf, pdf)

Review:

For dimensions of irreps

Textbook accounts:

  • Shlomo Sternberg, Section 5.4 and Appendix C.7 of: Group Theory and Physics, Cambridge University Press 1994

Review for Sym(n):

  • James Stevens, Section 2.2 of: Schur-Weyl duality (pdf)

Review for SU(n):

  • Sarah Peluse, Section 1 of: Irreducible representations of SU(n)SU(n) with prime power degree, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d (pdf)

  • Some Notes on Young Tableaux as useful for irreps of su(n)\mathrm{su}(n) (pdf)

  • Section 2 of: Group Theory primer SU(n)SU(n) (pdf)

Last revised on May 21, 2021 at 01:51:26. See the history of this page for a list of all contributions to it.