Thick Hook Decompositions in Young Diagrams

• Thick hook decompositions are controlled unions of diagonal hooks in a Young diagram that segment the partition into mesoscopic blocks for accurate combinatorial estimates.
• They provide a combinatorial framework to organize Naruse’s hook-length formula and derive uniform bounds on skew-tableau ratios and symmetric group characters.
• The method supports both representation theory for symmetric groups and applications in biparameter persistence by isolating local contributions via blockwise estimates.

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 $n$ (Teyssier, 9 Aug 2025).

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 $d_{\lambda \setminus \mu}$0 is the largest $d_{\lambda \setminus \mu}$1 such that $d_{\lambda \setminus \mu}$2 and $d_{\lambda \setminus \mu}$3. The hook at the $d_{\lambda \setminus \mu}$4th diagonal is the ordinary hook based at $d_{\lambda \setminus \mu}$5,

$d_{\lambda \setminus \mu}$6

More generally, for $d_{\lambda \setminus \mu}$7,

$d_{\lambda \setminus \mu}$8

and any such union is called a thick hook.

If $d_{\lambda \setminus \mu}$9, the skew shape $n^{2/3}$0 has $n^{2/3}$1 cells, and $n^{2/3}$2. An $n^{2/3}$3-thick-hook decomposition of $n^{2/3}$4 is a sequence

$n^{2/3}$5

with corresponding thick hooks

$n^{2/3}$6

such that

$n^{2/3}$7

the $n^{2/3}$8 are pairwise disjoint, and $n^{2/3}$9.

This definition isolates a mesoscopic scale between single hooks and the full diagram. The decomposition is designed to replace uncontrolled global dependence on $n$0 by blockwise estimates with cardinality bounds.

2. Existence and the combinatorial role of the decomposition

The existence statement in the exposition is explicit: if

$n$1

one can choose the decomposition greedily so that each $n$2 (Teyssier, 9 Aug 2025). This is the structural input behind the later bounds on excited-diagram sums. The factor $n$3 is the scale at which local contributions are controlled uniformly.

An illustrative example is given for

$n$4

for which $n$5, and with $n$6. The hooks at diagonal positions $n$7 are recorded as

$n$8

so that $n$9 is presented as an $\lambda \vdash n$0 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 $\lambda \vdash n$1, where $\lambda \vdash n$2 is the number of selected cells lying in $\lambda \vdash n$3. This reduces a geometric enumeration problem to a combination of elementary counting over compositions of $\lambda \vdash n$4 and local estimates on each block.

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

Fix $\lambda \vdash n$5 and $\lambda \vdash n$6 with $\lambda \vdash n$7. Naruse’s hook-length formula is expressed via

$\lambda \vdash n$8

and the falling factorial

$\lambda \vdash n$9

Then

$i$0

so that

$i$1

The key estimate concerns the horizontal strip $i$2. Whenever $i$3, for every $i$4 one has

$i$5

where $i$6 denotes the binomial coefficient (Teyssier, 9 Aug 2025). The proof idea is organized by an $i$7-decomposition $i$8. If an excited diagram $i$9 of $\lambda_i$0 contributes $\lambda_i$1 cells in $\lambda_i$2, with $\lambda_i$3, then the number of possible occupancy vectors is at most

$\lambda_i$4

Within each $\lambda_i$5, one shows a local bound

$\lambda_i$6

Multiplying these local factors yields the global estimate.

Two asymptotic regimes are then deduced. If $\lambda_i$7, then

$\lambda_i$8

If $\lambda_i$9, then

$u=(i,j)\in\lambda$0

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 $u=(i,j)\in\lambda$1, the exposition decomposes $u=(i,j)\in\lambda$2 into at most $u=(i,j)\in\lambda$3 “stair” lines $u=(i,j)\in\lambda$4, with

$u=(i,j)\in\lambda$5

This yields the multiplicative reduction

$u=(i,j)\in\lambda$6

Optimizing the resulting expression gives a universal constant $u=(i,j)\in\lambda$7 such that for any $u=(i,j)\in\lambda$8 and $u=(i,j)\in\lambda$9,

$$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$0

Since

$$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$1

division by $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$2 gives the stated bound on the skew-tableau ratio:

$$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$3

The exposition states that these bounds are sharp, up to constants, by taking $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$4 to be rectangles in various regimes (Teyssier, 9 Aug 2025).

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 $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$5, thereby connecting Naruse’s formula to uniform skew-dimension bounds.

5. Character bounds for symmetric groups

Let $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$6 have support size $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$7, meaning the number of nonfixed points. Write $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$8 for the fixed-point-free permutation in $$h_\lambda(u) = (\text{number of cells to the right of }u\text{ in row }i) + (\text{number of cells above }u\text{ in column }j) + 1.$$9 with the same nontrivial cycle lengths. The iterated Murnaghan–Nakayama expansion used in the exposition is

$$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$0

A separate Murnaghan–Nakayama-plus-triangle-inequality estimate gives, for any $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$1 with $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$2 cycles,

$$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$3

Since $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$4, one obtains in particular

$$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$5

Combining these character estimates with the skew-dimension bound and summing over the polynomially many $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$6 yields the uniform character bound

$$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$7

uniformly in $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$8 and $$d_\lambda := |SYT(\lambda)| = \frac{n!}{\prod_{u\in\lambda} h_\lambda(u)}.$$9 as $d_{\lambda \setminus \mu}$00 (Teyssier, 9 Aug 2025).

In the balanced case, if $d_{\lambda \setminus \mu}$01 is $d_{\lambda \setminus \mu}$02-balanced so that $d_{\lambda \setminus \mu}$03, this simplifies to

$d_{\lambda \setminus \mu}$04

The exposition states that this matches Moore–Russell’s conjecture in the regime $d_{\lambda \setminus \mu}$05, is new for $d_{\lambda \setminus \mu}$06, 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 $d_{\lambda \setminus \mu}$07 to be a long rectangle and $d_{\lambda \setminus \mu}$08 a single long cycle, one sees that the character bounds cannot be improved, up to the constant $d_{\lambda \setminus \mu}$09, in the various regimes of $d_{\lambda \setminus \mu}$10 versus $d_{\lambda \setminus \mu}$11 and $d_{\lambda \setminus \mu}$12 (Teyssier, 9 Aug 2025). 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 $d_{\lambda \setminus \mu}$13, showing that the $d_{\lambda \setminus \mu}$14-products of monoparameter modules are exactly the hook-decomposable modules and exactly the modules admitting a Smith-type structure theorem (Mastroianni et al., 24 Mar 2026). In that context, a hook summand is either a free summand $d_{\lambda \setminus \mu}$15 or a torsion summand

$d_{\lambda \setminus \mu}$16

and hook-decomposable modules have projective dimension $d_{\lambda \setminus \mu}$17 with explicit two-term resolutions.

That work also comments on possible “thick” hooks in a different sense: a region of $d_{\lambda \setminus \mu}$18 of the form

$d_{\lambda \setminus \mu}$19

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 $d_{\lambda \setminus \mu}$20, and admit no Smith-type classification; the missing corner breaks the tensor-product factorization (Mastroianni et al., 24 Mar 2026).

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.