---
title: Thick Hook Decompositions in Young Diagrams
url: https://www.emergentmind.com/topics/thick-hook-decompositions
type: topic
---

# Thick Hook Decompositions in Young Diagrams

Searching arXiv for the cited papers and nearby usage of "thick hook" terminology.
Thick hook decompositions are decompositions of a Young diagram into disjoint unions of diagonal hooks whose sizes are uniformly controlled. In the setting of partitions $\lambda \vdash n$, they provide a combinatorial device for organizing Naruse’s hook-length formula for skew tableaux and, through the Murnaghan–Nakayama rule, for deriving uniform bounds on irreducible characters of the symmetric group. In Teyssier’s formulation, the method yields bounds on $d_{\lambda \setminus \mu}$ for arbitrary skew shapes, improves the character bounds of Féray and Śniady for balanced representations when the support size is at least $n^{2/3}$, and is sharp for permutations with support size of order $n$ [2508.06770].

## 1. Basic objects and the definition of a thick hook

Let $\lambda \vdash n$ be a partition, written in French (matrix) convention, so that row $i$ has $\lambda_i$ boxes. For a cell $u=(i,j)\in\lambda$, the hook-length is
$$
h_\lambda(u)
=
(\text{number of cells to the right of }u\text{ in row }i)
+
(\text{number of cells above }u\text{ in column }j)
+
1.
$$
The classical hook-length formula gives
$$
d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.
$$

The diagonal length $\delta(\lambda)$ is the largest $k$ such that $\lambda_k\ge k$ and $\lambda'_k\ge k$. The hook at the $k$th diagonal is the ordinary hook based at $(k,k)$,
$$
\lambda^k
:=
\{\text{all cells in row }k\text{ from column }k\text{ to }\lambda_k\}
\cup
\{\text{all cells in column }k\text{ from row }k\text{ to }\lambda'_k\}.
$$
More generally, for $1\le j\le k\le \delta(\lambda)$,
$$
\lambda^{j\to k} := \bigcup_{i=j}^k \lambda^i,
$$
and any such union is called a thick hook.

If $\mu\subset \lambda$, the skew shape $\lambda\setminus\mu$ has $|\lambda\setminus\mu|=|\lambda|-|\mu|$ cells, and $d_{\lambda\setminus\mu}=|SYT(\lambda\setminus\mu)|$. An $(a,b)$-thick-hook decomposition of $\lambda$ is a sequence
$$
0=i_0<i_1<\cdots<i_p=\delta(\lambda),
$$
with corresponding thick hooks
$$
T_j := \lambda^{(i_{j-1}+1)\to i_j},\qquad j=1,\dots,p,
$$
such that
$$
a \le |T_j| \le b,
$$
the $T_j$ are pairwise disjoint, and $\bigcup_j T_j=\lambda$.

This definition isolates a mesoscopic scale between single hooks and the full diagram. The decomposition is designed to replace uncontrolled global dependence on $\lambda$ by blockwise estimates with cardinality bounds.

## 2. Existence and the combinatorial role of the decomposition

The existence statement in the exposition is explicit: if
$$
a \ge s(\lambda):=\max \text{ hook-length of }\lambda,
$$
one can choose the decomposition greedily so that each $|T_j|\in[a,4a]$ [2508.06770]. This is the structural input behind the later bounds on excited-diagram sums. The factor $4a$ is the scale at which local contributions are controlled uniformly.

An illustrative example is given for
$$
\lambda=[11,10,8,8,8,7,5,2,1],
$$
for which $\delta(\lambda)=5$, and with $a=8\ge s(\lambda)$. The hooks at diagonal positions $1,\dots,5$ are recorded as
$$
|\lambda^1|=34,\qquad |\lambda^2|=26,\qquad |\lambda^3\cup\lambda^4|=33,\qquad |\lambda^5|=23,
$$
so that $\{T_1,T_2,T_3,T_4\}$ is presented as an $(8,32)$ decomposition.

The significance of the construction is not merely partitioning the diagram. In subsequent estimates, every excited diagram is tracked through the vector of occupancies $(\ell_j)$, where $\ell_j$ is the number of selected cells lying in $T_j$. This reduces a geometric enumeration problem to a combination of elementary counting over compositions of $\ell$ and local estimates on each block.

## 3. Naruse’s hook-length formula and the excited-diagram estimate

Fix $\lambda\vdash n$ and $\mu\subset\lambda$ with $|\mu|=k$. Naruse’s hook-length formula is expressed via
$$
S(\lambda,\mu)
:=
\sum_{E\in Excited(\lambda,\mu)} \prod_{u\in E} h_\lambda(u),
$$
and the falling factorial
$$
n^{\downarrow k}=n(n-1)\cdots(n-k+1).
$$
Then
$$
d_{\lambda\setminus\mu}
=
d_\lambda\cdot \frac{S(\lambda,\mu)}{n^{\downarrow k}}
=
\frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}
\times
\frac{S(\lambda,\mu)}{n^{\downarrow k}},
$$
so that
$$
\frac{d_{\lambda\setminus\mu}}{d_\lambda}
=
\frac{S(\lambda,\mu)}{n^{\downarrow k}}.
$$

The key estimate concerns the horizontal strip $[\ell]$. Whenever $a\ge s(\lambda)$, for every $\ell\le \lambda_1$ one has
$$
S(\lambda,[\ell]) \le (4a)^\ell \cdot C(\ell+n/a,\ell),
$$
where $C(x,y)$ denotes the binomial coefficient [2508.06770]. The proof idea is organized by an $(a,4a)$-decomposition $T_1,\dots,T_p$. If an excited diagram $E$ of $[\ell]$ contributes $\ell_j$ cells in $T_j$, with $\sum \ell_j=\ell$, then the number of possible occupancy vectors is at most
$$
C(\ell+p,\ell)\le C(\ell+n/a,\ell).
$$
Within each $T_j$, one shows a local bound
$$
S^{*j}\le |T_j|^{\ell_j}\le (4a)^{\ell_j}.
$$
Multiplying these local factors yields the global estimate.

Two asymptotic regimes are then deduced. If $\ell\le n/s$, then
$$
S(\lambda,[\ell]) \le 8(n/\ell)^\ell.
$$
If $\ell\ge n/s$, then
$$
S(\lambda,[\ell]) \le 4e^2\, s^\ell.
$$
These are the basic line-shape estimates from which the general skew-shape bounds are assembled.

## 4. From line shapes to arbitrary skew shapes

For a general partition $\mu\vdash k$, the exposition decomposes $\mu$ into at most $2\delta(\mu)$ “stair” lines $\ell_1,\dots,\ell_q$, with
$$
q\le 2\sqrt{k}.
$$
This yields the multiplicative reduction
$$
S(\lambda,\mu)\le \prod_{i=1}^q S(\lambda,[\ell_i]).
$$
Optimizing the resulting expression gives a universal constant $C$ such that for any $\lambda\vdash n$ and $\mu\vdash k\subset\lambda$,
$$
S(\lambda,\mu)\le C^k\cdot \max\{s^k,(n/\sqrt{k})^k\}.
$$
Since
$$
n^{\downarrow k}\ge (n/e)^k,
$$
division by $n^{\downarrow k}$ gives the stated bound on the skew-tableau ratio:
$$
\frac{d_{\lambda\setminus\mu}}{d_\lambda}
\le
C\cdot \max\{1/\sqrt{k},(s(\lambda)/n)^k\}.
$$
The exposition states that these bounds are sharp, up to constants, by taking $\lambda,\mu$ to be rectangles in various regimes [2508.06770].

This chain of arguments is the central combinatorial contribution of the method. Thick hook decompositions enter at the level of the line-shape estimate, but the stairs decomposition promotes that estimate to arbitrary $\mu$, thereby connecting Naruse’s formula to uniform skew-dimension bounds.

## 5. Character bounds for symmetric groups

Let $\sigma\in S_n$ have support size $k$, meaning the number of nonfixed points. Write $\sigma^*$ for the fixed-point-free permutation in $S_k$ with the same nontrivial cycle lengths. The iterated Murnaghan–Nakayama expansion used in the exposition is
$$
\chi^\lambda(\sigma)
=
\sum_{\mu\subset\lambda,\ |\mu|=k}
\chi^\mu(\sigma^*)\cdot \frac{d_{\lambda\setminus\mu}}{d_\lambda}.
$$
A separate Murnaghan–Nakayama-plus-triangle-inequality estimate gives, for any $\pi\in S_m$ with $c$ cycles,
$$
|\chi^\mu(\pi)|\le 2^m\cdot \delta(\mu)^c.
$$
Since $\delta(\mu)\le \sqrt{m}$, one obtains in particular
$$
|\chi^\mu(\sigma^*)|\le 2^k\cdot k^{\#cycles(\sigma^*)}.
$$

Combining these character estimates with the skew-dimension bound and summing over the polynomially many $\mu$ yields the uniform character bound
$$
|\chi^\lambda(\sigma)|
\le
O(1)^k\cdot \frac{1}{(\sqrt{k})^k}\cdot
\max\left\{1,\left(\frac{s(\lambda)\sqrt{k}}{n}\right)^k\right\},
$$
uniformly in $\lambda\vdash n$ and $\sigma\ne id$ as $n\to\infty$ [2508.06770].

In the balanced case, if $\lambda$ is $C$-balanced so that $s(\lambda)=O(\sqrt{n})$, this simplifies to
$$
|\chi^\lambda(\sigma)|\le \frac{O(1)^k}{(\sqrt{k})^k},\qquad k=|\sigma|.
$$
The exposition states that this matches Moore–Russell’s conjecture in the regime $k\asymp n$, is new for $k\gg n^{2/3}$, and, together with the previous Féray–Śniady bounds, yields the best known uniform character estimates.

## 6. Sharpness, scope, and a distinct usage in persistence theory

The sharpness discussion is explicit. By choosing $\lambda$ to be a long rectangle and $\sigma$ a single long cycle, one sees that the character bounds cannot be improved, up to the constant $O(1)$, in the various regimes of $k$ versus $n$ and $s(\lambda)$ [2508.06770]. In that sense, the thick-hook method is not merely qualitative; it reaches the correct scale in the principal asymptotic regimes identified in the exposition.

A separate and terminologically distinct usage appears in biparameter persistence theory. Mastroianni, Guerra, Fugacci, and De Negri study hook-decomposable modules over $R=F[x,y]$, showing that the $\gamma$-products of monoparameter modules are exactly the hook-decomposable modules and exactly the modules admitting a Smith-type structure theorem [2603.23008]. In that context, a hook summand is either a free summand $R(-p)$ or a torsion summand
$$
(R/(x^{q_x-p_x}y^{q_y-p_y}))(-p),
$$
and hook-decomposable modules have projective dimension $1$ with explicit two-term resolutions.

That work also comments on possible “thick” hooks in a different sense: a region of $\mathbb{N}^2$ of the form
$$
\{\alpha\mid p\le \alpha\le r\}\setminus \{\alpha\mid q\le \alpha\le s\},
$$
described as a Young-diagram-shaped punctured rectangle. The conclusion there is negative: such thicker shapes are not in general cyclic, need not have projective dimension $1$, and admit no Smith-type classification; the missing corner breaks the tensor-product factorization [2603.23008].

A common source of ambiguity is therefore purely terminological. In algebraic combinatorics, thick hook decompositions refer to controlled unions of diagonal hooks inside a partition and are used analytically in Naruse–Murnaghan–Nakayama arguments. In biparameter persistence, “thick” hook language refers to a proposed extension beyond hook-decomposable modules and marks precisely the point where the Smith-type theory fails.

Source: https://www.emergentmind.com/topics/thick-hook-decompositions