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Hooks are bead-gap pairs on the abacus

Statement

Let $e \geq 2$ , let $λ ⊢ n$ and let $X = {h_{1} > h_{2} > \dots > h_{m}}$ be a $β$ -set for $λ$ , so that $P (X) = λ$ (Beta-sets, the shift, and the e-runner abacus of a partition). Write $H_{i} (λ)$ for the set of hook lengths in row $i$ of $[λ]$ : thus $h \in H_{i} (λ)$ means $h = h_{i, j} (λ)$ for some node $(i, j) \in [λ]$ , and $H_{i} (λ) = \emptyset$ for $i > ℓ (λ)$ (Hook, arm, leg, and hook length of a box).

(i) For every $h \geq 0$ and every $i \in [m]$ , $h \in H_{i} (λ)$ if and only if $h \leq h_{i}$ and $h_{i} - h \notin X$ . Equivalently, the hooks of $λ$ correspond bijectively to the bead-gap pairs $(b, g)$ with $b \in X$ , $g \notin X$ and $0 \leq g < b$ ; the hook $H_{i, j} (λ)$ corresponds to $(b, g) = (h_{i}, h_{i} - h_{i, j} (λ)),$ and then $∣ H_{i, j} (λ) ∣ = b - g$ .

(ii) Fix a node $(i, j) \in [λ]$ and let $(b, g)$ be its bead-gap pair as in (i). Then $X^{'} : = (X ∖ {b}) \cup {g}$ is again a $β$ -set, now for the partition $λ ∖ R_{i, j}$ obtained from $λ$ by removing the removable border strip $R_{i, j}$ that corresponds to the hook of $(i, j)$ , of size $h_{i, j} (λ)$ and height $ℓ (i, j) = λ_{j}^{'} - i$ (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips); in terms of parts, with $b_{0} : = ℓ (i, j) = λ_{j}^{'} - i$ , $λ ∖ R_{i, j} = (λ_{1}, \dots, λ_{i - 1}, λ_{i + 1} - 1, \dots, λ_{i + b_{0}} - 1, j - 1, λ_{i + b_{0} + 1}, \dots),$ so that $∣ λ ∖ R_{i, j} ∣ = n - h_{i, j} (λ)$ . Finally $X^{'}$ is obtained from the configuration $A_{X}$ by moving the bead from position $b$ to the gap $g$ .

(iii) In particular $e ∣ h_{i, j} (λ)$ if and only if $b \equiv g (m o d e)$ , that is, if and only if the bead $b$ and the gap $g$ lie on the same runner of the $e$ -abacus; then $g = b - e \cdot (h_{i, j} (λ) / e)$ and the move in (ii) is the slide of the bead $b$ up $h_{i, j} (λ) / e$ rows on its own runner. A hook of length $e$ in row $i$ of $λ$ exists if and only if $h_{i} \geq e$ and $h_{i} - e \notin X$ ; Thus $λ$ has a hook of length $e$ if and only if some bead of $X$ has a gap in the position $e$ labels above it on its own runner.

No choice principle is used.

Facts & Assumptions

Given: An integer $e \geq 2$ , a partition $λ ⊢ n$ with $t : = ℓ (λ)$ parts, and a $β$ -set $X = {h_{1} > \dots > h_{m}}$ for $λ$ , so that $P (X) = λ$ (Beta-sets, the shift, and the e-runner abacus of a partition).

[F1]

Hook lengths: $h_{i, j} (λ) = λ_{i} - j + λ_{j}^{'} - i + 1$ for $(i, j) \in [λ]$ , and for fixed $i$ the map $j \mapsto h_{i, j} (λ)$ is strictly decreasing on $1 \leq j \leq λ_{i}$ , because $h_{i, j} (λ) - h_{i, j + 1} (λ) = 1 + λ_{j}^{'} - λ_{j + 1}^{'} \geq 1$ ; hence the hook lengths in a fixed row are pairwise distinct and $h_{i, j} (λ) \geq 1$ (Hook, arm, leg, and hook length of a box).

[F2]

For a $β$ -set $Y = {y_{1} > \dots > y_{r}}$ one has $P (Y)_{j} = y_{j} - (r - j)$ for $1 \leq j \leq r$ , after which trailing zeros are deleted: indeed there are $y_{j}$ positions $0, 1, \dots, y_{j} - 1$ below the bead $y_{j}$ , of which exactly $r - j$ are occupied by the beads $y_{j + 1}, \dots, y_{r}$ , so $y_{j} - (r - j)$ of them are gaps, and the definition of $P$ takes exactly this difference (Beta-sets, the shift, and the e-runner abacus of a partition).

[F3]

The $β$ -sets of $λ$ are exactly the sets $X_{λ} + l$ , $l \in N_{0}$ , where $X_{λ} = {b_{1} (λ) > \dots > b_{t} (λ)}$ with $b_{i} (λ) = h_{i, 1} (λ) = λ_{i} + t - i$ , and $∣ X_{λ} + l ∣ = t + l$ (Beta-sets, the shift, and the e-runner abacus of a partition).

[F4]

A removable border strip of $λ$ is a nonempty edge-connected set of nodes containing no $2 \times 2$ block whose complement in $[λ]$ is again a Young diagram; its height is its number of occupied rows minus one, and for a strip that is the hook of a node it equals the leg length of that node (Removable border strips and their height, Hooks of a partition correspond bijectively to removable border strips).

Proof

technique · direct

1.1F1F2F3algebra

Pad $λ$ with zeros to length $m$ . By [F3], $h_{i} = λ_{i} + m - i$ . Below $h_{i}$ there are exactly $m - i$ beads, hence $h_{i} - (m - i) = λ_{i}$ gaps. For a node $(i, j)$ put $g_{j} = h_{i} - h_{i, j} (λ) = m + j - 1 - λ_{j}^{'}$ . These values are distinct by [F1], satisfy $0 \leq g_{j} < h_{i}$ , and are gaps: equality $g_{j} = h_{x} = λ_{x} + m - x$ would give $λ_{x} - j + λ_{j}^{'} - x + 1 = 0$ . If $x \leq λ_{j}^{'}$ , both differences are nonnegative; otherwise both are at most $- 1$ , since $λ_{x} < j$ . Neither case gives zero. Thus the $λ_{i}$ numbers $g_{j}$ exhaust the gaps below $h_{i}$ , including the empty set when $λ_{i} = 0$ .

1.2F1F4constructalgebra

Fix $(i, j)$ and put $r = λ_{j}^{'}$ and $g = m + j - 1 - r$ . The rim strip of [F4] has complement with rows $μ_{p} = λ_{p}$ outside $i \leq p \leq r$ , $μ_{p} = λ_{p + 1} - 1$ for $i \leq p < r$ , and $μ_{r} = j - 1$ : in row $p$ the strip occupies the rim nodes with column at least $j$ , whose complement ends at $\max (j - 1, λ_{p + 1} - 1)$ . For $p < r$ one has $λ_{p + 1} \geq j$ , while $λ_{r + 1} < j$ . These formulas give the displayed partition in (ii), with trailing zeros deleted, and its size is $n - h_{i, j} (λ)$ .

2.1F1step 1.1algebra

Consequently, for $h \geq 0$ , $h \in H_{i} (λ)$ exactly when $h \leq h_{i}$ and $h_{i} - h \notin X$ : at $h = 0$ the latter fails because $h_{i} \in X$ , and for $h > h_{i}$ its first inequality fails. For $1 \leq h \leq h_{i}$ this is the gap correspondence in step 1.1. The bead $h_{i}$ determines $i$ , and the distinct row hook lengths determine $j$ , so $(i, j) \mapsto (h_{i}, g_{j})$ is the asserted bijection with length $h_{i} - g_{j}$ .

2.2F2step 1.1step 1.2algebra

The new value $g$ is below $h_{r}$ because $h_{r} - g = λ_{r} - j + 1 \geq 1$ , and, if $r < m$ , above $h_{r + 1}$ because $g - h_{r + 1} = j - λ_{r + 1} \geq 1$ . Hence the decreasing list of $X^{'} = (X ∖ {h_{i}}) \cup {g}$ is $h_{1}, \dots, h_{i - 1}, h_{i + 1}, \dots, h_{r}, g, h_{r + 1}, \dots, h_{m}$ . Applying [F2] gives precisely the parts $μ_{p}$ in step 1.2. Thus $P (X^{'}) = λ ∖ R_{i, j}$ , and the abacus change is the relocation of the bead $h_{i}$ to $g$ .

3.1step 2.1step 2.2algebra∎

The length is $b - g$ , so $e ∣ h_{i, j} (λ)$ exactly when $b \equiv g (m o d e)$ . In that case the bead remains on its runner, moves up $h_{i, j} (λ) / e$ rows, and all runner bead counts are unchanged. Without this divisibility assumption the relocation may change runners. In row $i$ a hook of length $e$ exists exactly when $h_{i} \geq e$ and $h_{i} - e \notin X$ , by step 2.1; existential quantification over $i$ gives the final bead criterion in (iii).

Depends on

Used by

The e-core, the e-quotient, and the e-weight Definition
The three-core and three-quotient of (6,5,2,1³) by abacus Example
The three-core of (4,3,2,1) by two removal orders Example
The e-core is independent of the rim hook removal order Theorem
The e-quotient controls the hooks of e-divisible length Theorem

Dependency tree · two levels

8 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

Stacey Law (lecturer), notes by Leonard Tomczak, Representation Theory of Symmetric Groups, Cambridge Part III Michaelmas 2022 lecture notes, complete 76-page PDF (standard reference, not scraped)
Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete 79-page prepublication manuscript (1998; expanded AMS University Lecture Series 15 edition, 1999) (standard reference, not scraped)