# nLab hook-content formula

Contents

### Context

#### Combinatorics

combinatorics

enumerative combinatorics

graph theory

rewriting

### Polytopes

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category: combinatorics

# Contents

## Idea

The “hook-content formula” expresses the number of semistandard Young tableaux with fixed underlying Young diagram and upper bound on its labels in terms of the “hook lengths” and the “content of boxes” of the underlying diagram.

If the dependency in the “contents” is removed, the formula reduces to the hook length formula that counts standard Young tableaux of the given shape.

Both of these formulas equivalently give dimensions of irreps in representation theory:

See at hook length formula for more on this.

## 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 “$\ell hook$” 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

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

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

Alongside this terminology one says that

$content(i,j) \;\coloneqq\; j - i \;\; \in \mathbb{Z}$

is the content of the box $(i,j)$.

Consider in the following:

• $n \in \mathbb{N}$ a natural number;

• $\lambda = (\lambda_1 \geq \cdots \geq \lambda_{rows(\lambda)})$ a partition of $n$, $\underset{i}{\sum} \lambda_i = n$,

equivalently a Young diagram with $n$ boxes;

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

Write:

• $ssYT_\lambda(N)$ for the set of semistandard Young tableaux $T$

• of shape (i.e. with underlying Young diagram) $\lambda$,

• with labels $T_{i,j} \leq N$ (i.e. with $T_{i,j} \in \{1, \cdots, N\}$).

• $n(T) = \sum_{i,j} T_{i,j}$ for the sum of all the labels.

## Statement

### Standard hook-content formula

###### Proposition

For $\lambda$ a partition/Young diagram and $N \in \mathbb{N}_+$, the number $\left\vert ssYTableaux_\lambda(N)\right\vert$ of semistandard Young tableaux $T$ of shape $\lambda$ (hence the value of the Schur polynomial $s_\lambda$ on $N$ unit argument) and entries bounded by $T_{i,j} \leq N$ is:

$\left\vert ssYTableaux_\lambda(N)\right\vert \;=\; s_{\lambda} \big( x_1 \!=\! 1, \cdots, x_N \!=\! 1 \big) \;\; = \;\; \underset{ (i,j) }{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) } \,.$

### $q$-Deformed hook-content formula

With the above ingredients, we have the following equalities of polynomials in a variable $q$:

$\underset{ T \in ssYT_\lambda(N) }{\sum} q^{ n(T) } \;\;=\;\; q^{ \sum_i i \cdot \lambda_i } \cdot \!\!\!\!\!\!\! \underset{ { i \in \{1, \cdots, rows(\lambda)\} } \atop {j \in \{1, \cdots, \lambda_j\}} }{\prod} \frac{ 1 - q^{N + content(i,j)} }{ 1 - q^{\ell hook(i,j)} }$

### Standard

The original statement and 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: