nLab hook-content formula

For disambiguation of content see there.

Idea

In combinatorics and representation theory, 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:

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)

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:

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

(Stanley 71, Thm. 15.3, Stanley 99, Thm. 7.21.2)

$q$-Deformed hook-content formula

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

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

(Krattenthaler 98, Thm. 1)

Related concepts

• hook length formula

• Plancherel measure, Schur-Weyl measure

References

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:

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

See also:

• Graham Hawkes, An Elementary Proof of the Hook Content Formula (arXiv:1310.5919)

Generalized

• Christian Krattenthaler, An Involution Principle-Free Bijective Proof of Stanley’s Hook-Content Formula, Discrete Mathematics and Theoretical Computer Science 3, 1998, 11–32 (hal:hal-00958904, pdf)

• Christian Krattenthaler, Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A 88 (1999), 66-92 (arXiv:math/9807068)

See also:

• Mark Wildon, A corollary of Stanley’s Hook Content Formula (arXiv:1904.08904)