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Nakayama block classification for symmetric groups

Statement

Let $k$ be a splitting field of characteristic $p > 0$ (equivalently the residue field of the fixed splitting $p$ -modular system, A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras, p-blocks from primitive central idempotents) and let $n \geq 0$ . Two Specht modules $S_{k}^{λ}, S_{k}^{μ}$ of $k [S_{n}]$ lie in the same block of $k [S_{n}]$ if and only if $λ, μ ⊢ n$ have the same $p$ -core (The e-core, the e-quotient, and the e-weight). Consequently the blocks of $k [S_{n}]$ are indexed by the pairs $(γ, w)$ with $γ$ a $p$ -core, $w \in N_{0}$ and $n = ∣ γ ∣ + p w$ , and the block of $S_{k}^{λ}$ is the pair $(C_{p} (λ), w_{p} (λ))$ with $w_{p} (λ) = (n - ∣ C_{p} (λ) ∣) / p$ (The core-quotient size formula defines the weight).

No choice principle is used.

Facts & Assumptions

Given: A splitting field $k$ of characteristic $p > 0$ with the fixed splitting $p$ -modular system, an integer $n \geq 0$ , and the group algebra $k [S_{n}]$ with group Specht modules $S_{k}^{λ}$ for $λ ⊢ n$ .

[F1]

The algebra isomorphism $H_{k, 1} (S_{n}) ≅ k [S_{n}]$ sends each ordinary Dipper-James Specht module to the classical group Specht module of the same shape, and the quantum characteristic at $q = 1$ in characteristic $p$ equals $p$ (Hecke Specht theory specialises to group Specht theory at q = 1, The quantum characteristic of an invertible Hecke parameter). The module comparison uses the accepted signed conjugate Murphy/ordinary dictionary before specialising; it does not identify the same-shape Murphy cell module with the ordinary Specht module.

[F2]

The Hecke block criterion, in the Dipper-James normalisation of the same Specht modules as [F1]: Type-A Hecke Specht modules are in the same block iff their labels have the same e-core states that for a field $k$ and $q \in k^{\times}$ of quantum characteristic $e \geq 2$ , two Specht modules $S^{λ}, S^{μ}$ of $H_{k, q} (S_{n})$ lie in the same block if and only if $C_{e} (λ) = C_{e} (μ)$ , and that the blocks are indexed by the $e$ -cores $γ$ with $∣ γ ∣ \leq n$ and $∣ γ ∣ \equiv n (m o d e)$ .

[F4]

The core-quotient size formula The core-quotient size formula defines the weight: for every $e \geq 2$ and $λ ⊢ n$ one has $n = ∣ C_{e} (λ) ∣ + e w_{e} (λ)$ with $w_{e} (λ) = ∣ Q_{e} (λ) ∣ \in N_{0}$ , and for an $e$ -core $γ$ the condition $∣ γ ∣ \leq n$ , $∣ γ ∣ \equiv n (m o d e)$ is exactly the condition that $n - ∣ γ ∣$ be an admissible $e$ -weight; conversely every such $γ$ occurs as the core of a partition of $n$ (q-Schur Weyl modules are in the same block iff their labels have the same e-core, The e-core is independent of the rim hook removal order).

[F5]

An algebra isomorphism $φ : A \to B$ preserves blocks: it maps a central idempotent $z$ to the central idempotent $φ (z)$ , preserves primitivity, and hence induces a bijection between the block sets (Blocks of a finite-dimensional algebra via primitive central idempotents, Decompositions of the identity into primitive central idempotents are unique).

Proof

technique · direct

1.1F1F5algebra

Specialisation. Take $q = 1 \in k^{\times}$ and $e$ the quantum characteristic of $(k, q)$ ; since $q = 1$ and $char k = p > 0$ , [F1] gives $e = p$ . By [F1] there is an algebra isomorphism $H_{k, 1} (S_{n}) ≅ k [S_{n}]$ carrying the Hecke Specht modules of $H_{k, 1} (S_{n})$ to the group Specht modules $S_{k}^{λ}$ , so by [F5] this isomorphism induces a bijection between the blocks of the two algebras that matches $S^{λ}$ with $S_{k}^{λ}$ ; in particular $S_{k}^{λ}$ and $S_{k}^{μ}$ lie in the same block of $k [S_{n}]$ if and only if the corresponding Hecke Specht modules lie in the same block of $H_{k, 1} (S_{n})$ .

2.1F2step 1.1

The core criterion. Applying the Hecke block criterion [F2] with $q = 1$ and $e = p$ , the Hecke Specht modules of $H_{k, 1} (S_{n})$ lie in the same block if and only if the labels $λ, μ$ have the same $p$ -core. Together with step 1.1 this gives: $S_{k}^{λ}$ and $S_{k}^{μ}$ lie in the same block of $k [S_{n}]$ if and only if $C_{p} (λ) = C_{p} (μ)$ .

3.1F4step 2.1algebra

The parametrisation by cores and weights. By [F2] the blocks of $H_{k, 1} (S_{n})$ , hence by step 1.1 the blocks of $k [S_{n}]$ , are indexed by the $p$ -cores $γ$ with $∣ γ ∣ \leq n$ and $∣ γ ∣ \equiv n (m o d p)$ ; the block of $S_{k}^{λ}$ corresponds to $C_{p} (λ)$ . For $n \geq 0$ and such a $γ$ put $w : = (n - ∣ γ ∣) / p \in N_{0}$ ; by [F4] this is exactly the $p$ -weight $w_{p} (λ)$ of any Specht label with core $γ$ , and $n = ∣ γ ∣ + p w$ . Since $w$ is determined by $n$ and $γ$ , the map $γ \mapsto (γ, w)$ is a bijection onto the displayed set of pairs, and the block of $S_{k}^{λ}$ is the pair $(C_{p} (λ), w_{p} (λ))$ .

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 establish both assertions: the same-block criterion by the $p$ -core, and the indexing of the blocks by the pairs $(γ, w)$ with $n = ∣ γ ∣ + p w$ together with the identification of the block of $S_{k}^{λ}$ as $(C_{p} (λ), w_{p} (λ))$ .

The classification is stated independently, by a Young-module and Brauer-quotient route, as Theorem 1 of O'Donovan's preprint recorded in the sources; that reference is an independent confirmation of the statement and is not used in the proof above. No choice principle is used anywhere in this item.

Depends on

Used by

Equal removable-hook counts do not replace the core test Counterexample
The complete weight-one decomposition matrix for S3 at p=3 Example
Weight-zero symmetric-group blocks are defect zero Proposition

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Sources

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)
Joseph O'Donovan, On the blocks of the symmetric group, arXiv:1708.04365 (complete preprint) (standard reference, not scraped)