nLab hook length formula

Idea

Given a partition/Young diagram $\lambda$ with $n$ boxes/of $n$:

the hook length formula expresses both

• the number of standard Young tableaux of shape $\lambda$,

• the dimension of the irreducible representation of Sym(n) labelled by $\lambda$ (i.e. of the Specht module $S^{(\lambda)}$)

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

similarly, the hook-content formula expresses both

• the number of semistandard Young tableau of shape $\lambda$;

• the dimension of the irreducible representation of SU(n)

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

hook length formula	hook-content formula
number of standard Young tableaux	number 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 “$\ell hook_\lambda$” for the length of such a hook, i.e. for the number of boxes it contains. Formally:

Definition

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

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

• $j \in \{1, \cdots, \lambda_i\}$

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

$$\ell hook_\lambda(i,j) \;\coloneqq\; 1 + (\lambda_i - j) + (\lambda'_j - i) \,,$$

where $\lambda'$ denotes the conjugate partition (see there). The quantities $\lambda_i - j$ and $\lambda'_j - i$ are commonly referred to as the arm length and leg length of box $(i,j)$, respectively.

Definition

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

• the number of standard Young tableaux:

  (1)$$\left\vert sYTableaux_\lambda\right\vert \;\in\; \mathbb{N}_+$$

• the number of standard Young tableaux with bounded entries $T_{i j} \leq N$:

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

of shape $\lambda$.

Details

Counting standard Young tableaux

Proposition

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

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

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 $Sym(n)$

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

$$dim(S^{(\lambda)}) \;=\; \left\vert sYTableaux_\lambda \right\vert$$

(e.g. Sagan, Thm. 2.6.5)

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

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

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

• a partition (Young diagram) $\lambda$ of $n$ (boxes),

• a positive natural number $N \in \mathbb{N}_+$,

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

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

Here

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

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

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

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

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

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

This appears as Sternberg 94 (C.27)

Related concepts

• hook-content formula

• Plancherel measure

• Schur-Weyl measure

References

For standard Young tableaux

The original proof is due to:

• James Sutherland Frame, Gilbert de Beauregard Robinson, Robert McDowell Thrall, The Hook Graphs of the Symmetric Group, Canadian Journal of Mathematics, Volume 6, 1954, pp. 316 - 324 (doi:10.4153/CJM-1954-030-1)

Textbook accounts:

• G. D. James, Thm. 20.1 in: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, volume 682, Springer 1978 (doi:10.1007/BFb0067708, pdf)

• Richard Stanley, Cor. 7.21.6 in: Enumerative combinatorics 2, Cambridge University Press (1999, 2010) (doi:10.1017/CBO9780511609589, webpage)

• Bruce Sagan, Thm. 3.10.2 & Thm. 2.6.5 in: The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)

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:

• Wikipedia, Hook length formula

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:

• Richard Stanley, Thm. 7.21.2 in: Enumerative combinatorics 2, Cambridge University Press (1999, 2010) (doi:10.1017/CBO9780511609589, webpage)

• A proof of the HCF (pdf, pdf)

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 $SU(n)$ with prime power degree, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d (pdf)

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

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