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p-regular and p-restricted partitions

Definition

Let $p$ be a prime and let $λ ⊢ n$ be a partition of $n \geq 0$ with parts $λ_{1} \geq λ_{2} \geq \dots \geq λ_{k} \geq 1$ and conjugate partition $λ^{'}$ , whose $j$ -th part is the number $λ_{j}^{'} = # {i : λ_{i} \geq j}$ of nodes of $[λ]$ in column $j$ (Partitions, English diagrams, and conjugation). For $j \geq 1$ let $z_{j} (λ) : = # {i : λ_{i} = j}$ be the multiplicity with which the positive integer $j$ occurs as a part of $λ$ ; only finitely many $z_{j}$ are nonzero.

$λ$ is $p$ -regular when every positive part of $λ$ occurs fewer than $p$ times, that is, when $z_{j} (λ) < p$ for every $j \geq 1$ .
$λ$ is $p$ -restricted when $λ_{i} - λ_{i + 1} < p$ for every $i \geq 1$ , where the sequence is padded by the trailing zeros $λ_{i} : = 0$ for $i > k$ .

Both conditions are finite families of inequalities. For $i > k$ one has $λ_{i} - λ_{i + 1} = 0 < p$ , so the restrictedness condition is the finite list for $1 \leq i \leq k$ . The empty partition $\emptyset ⊢ 0$ has no parts and $k = 0$ , so $z_{j} (\emptyset) = 0 < p$ and $0 - 0 = 0 < p$ for every $j, i \geq 1$ : it is simultaneously $p$ -regular and $p$ -restricted for every prime $p$ .

The two conditions are exchanged by conjugation: $λ$ is $p$ -restricted if and only if $λ^{'}$ is $p$ -regular. The number of columns of $[λ]$ of height exactly $i$ is the difference $λ_{i} - λ_{i + 1}$ of consecutive parts, and these heights are precisely the parts of $λ^{'}$ , so the multiplicity of the part $i$ in $λ^{'}$ is $λ_{i} - λ_{i + 1}$ ; the stated equivalence compares the same integers with $p$ .

The names record two genuinely different label conventions used later on this page: James's modular simple modules $D^{λ}$ are labelled by $p$ -regular $λ$ . Dual Specht modules themselves are defined for every partition; their simple heads give the $p$ -restricted labelling of simple modules, related to the first convention by transposition and a sign twist. Neither class of partitions contains the other in general.

Facts & Assumptions

Given: A prime $p$ and a partition $λ ⊢ n$ with $k$ parts and conjugate $λ^{'}$ .

[F1]

A partition of $n$ is a finite weakly decreasing sequence $λ = (λ_{1}, \dots, λ_{k})$ of positive integers with sum $n$ ; trailing zeros are not parts. Its conjugate has parts $λ_{j}^{'} = # {i : λ_{i} \geq j}$ and is again a partition of $n$ (Partitions, English diagrams, and conjugation).

[F2]

Column $j$ of $[λ]$ carries exactly $λ_{j}^{'}$ nodes, and row $i$ carries exactly $λ_{i}$ nodes (Partitions, English diagrams, and conjugation).

Proof

technique · direct

1.1givenF1F2algebra

Since the parts of $λ$ are weakly decreasing, the rows of length at least $i$ are exactly the first $λ_{i}^{'}$ rows, so the rows of length exactly $i$ are rows $λ_{i + 1}^{'} + 1, \dots, λ_{i}^{'}$ and there are $λ_{i}^{'} - λ_{i + 1}^{'}$ of them.

1.2givenF1F2algebra

Equivalently, the columns of height exactly $i$ number $λ_{i} - λ_{i + 1}$ : column $j$ has height $# {r : λ_{r} \geq j}$ , which is at least $i$ exactly when $j \leq λ_{i}$ , so the columns of height at least $i$ are columns $1, \dots, λ_{i}$ and those of height exactly $i$ number $λ_{i} - λ_{i + 1}$ .

2.1givenF1step 1.2algebra

By [F1] the parts of $λ^{'}$ are the column heights of $[λ]$ , so the multiplicity of the part $i$ in $λ^{'}$ is the number of columns of height exactly $i$ , namely $λ_{i} - λ_{i + 1}$ by step 1.2. Hence $λ^{'}$ is $p$ -regular if and only if $λ_{i} - λ_{i + 1} < p$ for every $i \geq 1$ , which is exactly the statement that $λ$ is $p$ -restricted.

2.2givenF1step 1.1step 1.2

The conjugation statement includes $n = 0$ : for $λ = \emptyset$ the conjugate is $\emptyset$ by [F1], both defining conditions are the empty family of inequalities, and steps 1.1-1.2 give $λ_{i} - λ_{i + 1} = 0$ for every $i$ .

3.1givenF1step 1.2step 2.1∎

Taking $i > k$ in the definition gives $λ_{i} - λ_{i + 1} = 0 < p$ , so the restrictedness condition is finite and the displayed equivalence of step 2.1 is a comparison of the same integers $z_{i} (λ^{'}) = λ_{i} - λ_{i + 1}$ with $p$ ; this proves the asserted conjugation statement.

Depends on

Partitions, English diagrams, and conjugation

Used by

p-regular and p-restricted labels differ Counterexample
Decomposition matrices of S3 at p=2 and p=3 Example
Nonzero maps into tabloid quotients force dominance Lemma
Specht Gram gcd detects p-regularity Lemma
p-regular and p-restricted labels under transpose and sign Proposition
Triangularity does not compute every modular decomposition number Remark
Dominance unitriangularity of the symmetric-group decomposition matrix Theorem
Modular simple modules of the symmetric group Theorem
Nonzero modular Specht quotient criterion Theorem

Dependency tree · two levels

2 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

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, §10.1 definition and Lemma 10.2, printed pp. 36-37 (standard reference, not scraped)
David A. Craven, Groups, Geometries and Representation Theory, §2.3, printed pp. 23-24 (p-regular partitions and the reversed-row construction) (standard reference, not scraped)