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The hook length formula

Statement

For every $n \geq 0$ and every $λ ⊢ n$ , the number $f^{λ}$ of standard $λ$ -tableaux is $f^{λ} = \frac{n!}{\prod_{x \in [λ]} h (x)},$ the empty product for $λ = \emptyset$ being $1$ , so that $f^{\emptyset} = 1$ and, for $n \geq 1$ , $f^{λ} = 1$ for $λ = (n)$ and $λ = (1^{n})$ . In particular, over $C$ , $\dim_{C} S^{λ} = \frac{n!}{\prod_{x \in [λ]} h (x)}$ for the Specht module $S^{λ}$ , including $\dim_{C} S^{\emptyset} = 1$ .

Facts & Assumptions

Given: An integer $n \geq 0$ and a partition $λ ⊢ n$ , with $f^{λ}$ the number of standard $λ$ -tableaux and $P (λ) = \prod_{x \in [λ]} h (x)$ the hook product.

[F1]

$f^{λ} = \sum_{x \in Rem (λ)} f^{λ - x}$ for $n \geq 1$ , and $f^{\emptyset} = 1$ ; the boxes of $[λ]$ are the boxes of $[λ - x]$ together with $x$ for $x \in Rem (λ)$ (The removal recursion for standard tableaux).

[F2]

For $x \in Rem (λ)$ with $λ ⊢ n \geq 1$ : $P (λ) / P (λ - x) = R (x) : = \prod_{y \in R_{x}} h_{λ} (y) / (h_{λ} (y) - 1)$ (Removing a corner changes hooks in its row and column).

[F3]

$\sum_{x \in Rem (λ)} R (x) = n$ , the empty sum being $0$ (The hook-product ratios sum to the size).

[F4]

The family of standard polytabloids is a $C$ -basis of the Specht module $S^{λ}$ , so $\dim_{C} S^{λ} = f^{λ}$ for every $λ ⊢ n$ , including $n = 0$ (Standard polytabloids form a basis of a complex Specht module).

[F5]

$P (λ)$ is the product of the $n$ positive integers $h (x)$ , one for each box of $[λ]$ ; for $λ = \emptyset$ it is the empty product $1$ , for $λ = (n)$ the product is $n!$ , and for $λ = (1^{n})$ the conjugate diagram gives the same multiset of hooks (Hook, arm, leg, and hook length of a box).

Proof

technique · strong induction on $n$

1.1baseF1F5given

Base cases: for $n = 0$ the only partition is $\emptyset$ , whose set of standard tableaux is the singleton consisting of the empty tableau, so $f^{\emptyset} = 1 = 0! / 1$ with empty product $1$ ; for $n = 1$ the only partition is $(1)$ , whose single box has $h = 1$ and exactly one standard tableau, so $f^{(1)} = 1 = 1! / 1$ .

1.2ihgiven

Induction hypothesis: for every $m$ with $0 \leq m < n$ and every partition $μ ⊢ m$ , $f^{μ} = m! / P (μ)$ .

1.3F4given

The dimension clause: by [F4], $\dim_{C} S^{λ} = f^{λ}$ for every $λ ⊢ n$ , including $λ = \emptyset$ where both sides are $1$ ; this holds for all $n$ because [F4] covers every $n \geq 0$ .

2.1step 1.2F1F2F3algebra

For $n \geq 2$ and $λ ⊢ n$ , [F1] gives $f^{λ} = \sum_{x \in Rem (λ)} f^{λ - x}$ ; each $λ - x$ is a partition of $n - 1 < n$ , so step 1.2 gives $f^{λ - x} = (n - 1)! / P (λ - x)$ , and [F2] turns this into $(n - 1)! R (x) / P (λ)$ . Summing over the removable nodes and using [F3], $f^{λ} = \frac{(n - 1)!}{P (λ)} \sum_{x} R (x) = \frac{(n - 1)! n}{P (λ)} = \frac{n!}{P (λ)}$ .

3.1step 2.1F5given

The two identities $f^{(n)} = 1 = f^{(1^{n})}$ follow because the hook multiset of $(n)$ and of $(1^{n})$ is ${1, 2, \dots, n}$ by [F5], so the formula gives $n! / n! = 1$ in both cases.

4.1step 1.1step 1.2step 2.1step 1.3step 3.1discharge-induction∎

Strong induction on $n$ : the base cases are step 1.1, the inductive step is step 2.1 with the hypothesis step 1.2, and steps 1.3 and 3.1 record the dimension and endpoint clauses; hence $f^{λ} = n! / \prod_{x \in [λ]} h (x)$ and $\dim_{C} S^{λ} = f^{λ}$ hold for every $n \geq 0$ and every $λ ⊢ n$ .

Depends on

Used by

Empty and singleton RSK boundaries Example
Hook lengths for one-row, one-column and hook shapes Example
Hook table for the shape (3,2,1) Example

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David A. Craven, Groups, Geometries and Representation Theory (Spring Term 2013 lecture notes, 42 pp.) (standard reference, not scraped)
Charlotte Chan, Representation Theory of Symmetric Groups (Oxford Hilary Term 2011 lecture notes, 40 PDF pp.) (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, MIT 18.712 Chapter 4 (OCW Chapter 4 file, 32 pp.) (standard reference, not scraped)