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Weight-zero symmetric-group blocks are defect zero

Statement

Let $k$ be a splitting field of characteristic $p > 0$ and let $λ ⊢ n$ be a partition of $p$ -weight $w_{p} (λ) = 0$ , that is, a $p$ -core (The e-core, the e-quotient, and the e-weight). Then the block $B$ of $k [S_{n}]$ containing $S_{k}^{λ}$ (Nakayama block classification for symmetric groups) contains exactly one Specht label, namely $λ$ ; its decomposition matrix is the $1 \times 1$ matrix $(1)$ , so $D^{λ}$ is both the unique simple and the unique projective indecomposable module of $B$ ; its Cartan matrix is $(1)$ ; the block algebra is a full matrix algebra over $k$ ; and $B$ has trivial defect group, that is, defect zero in the fixed splitting modular system (Defect group and numerical defect of a block).

No choice principle is used.

Facts & Assumptions

Given: A splitting field $k$ of characteristic $p > 0$ for $S_{n}$ and all its subgroups, an integer $n \geq 0$ , a partition $λ ⊢ n$ with $w_{p} (λ) = 0$ , and the block $B$ of $k [S_{n}]$ containing $S_{k}^{λ}$ .

[F1]

$w_{p} (λ) = (n - ∣ C_{p} (λ) ∣) / p$ , so $w_{p} (λ) = 0$ is equivalent to $C_{p} (λ) = λ$ , and a $p$ -core is $p$ -regular (The e-core, the e-quotient, and the e-weight, The core-quotient size formula defines the weight). Hence $D^{λ} \neq 0$ and $S_{k}^{λ}$ has simple head $D^{λ}$ , with $D^{λ}$ its unique maximal quotient, so $S_{k}^{λ}$ is indecomposable (Modular simple modules of the symmetric group).

[F2]

Nakayama: two Specht modules of $k [S_{n}]$ lie in the same block if and only if their labels have the same $p$ -core (Nakayama block classification for symmetric groups); the blocks are indexed by the $p$ -cores $γ$ with $∣ γ ∣ \leq n$ , $∣ γ ∣ \equiv n (m o d p)$ . The simple $k [S_{n}]$ -modules are exactly the $D^{μ}$ for the $p$ -regular $μ ⊢ n$ , pairwise non-isomorphic (Modular simple modules of the symmetric group).

[F3]

Block bookkeeping: the identity of $k [S_{n}]$ decomposes uniquely into primitive central idempotents, every finite-dimensional module decomposes along the blocks, and a simple module lies in exactly one block (p-blocks from primitive central idempotents, Blocks of a finite-dimensional algebra via primitive central idempotents); in particular an indecomposable finite-dimensional module lies in a single block.

[F4]

Dominance unitriangularity: $d_{μ μ} = 1$ and $d_{μ ν} \neq 0$ implies $ν ⊵ μ$ for the modular decomposition numbers $d_{μ ν} = [S_{k}^{μ} : D^{ν}]$ (Dominance unitriangularity of the symmetric-group decomposition matrix).

[F5]

For a block $B$ with decomposition matrix $D_{B}$ and Cartan matrix $C_{B}$ one has $C_{B} = D_{B}^{T} D_{B}$ (The Cartan matrix is D^T D); the $(1, 1)$ -entry of $C_{B}$ is the multiplicity of the simple module in its projective cover (Projective indecomposable characters and Cartan invariants).

[F6]

Defect-zero criterion: over the splitting residue field, for a block $B$ the following are equivalent: the defect group of $B$ is trivial; $B$ is a full matrix algebra over $k$ ; $B$ has a projective simple module; in that case there is exactly one simple module and it is projective (Defect zero blocks are simple algebras).

Proof

technique · direct

1.1F1F2F3F4

Bookkeeping for the labels of $B$ . Let $ν ⊢ n$ be a $p$ -regular partition with block $B (ν) = B$ ; by [F1] and [F2] this is equivalent to $C_{p} (ν) = C_{p} (λ) = λ$ , and since $∣ ν ∣ = n = ∣ λ ∣$ this forces $ν = λ$ . Hence the only $p$ -regular label in $B$ is $λ$ , and by [F2] the only simple module in $B$ is $D^{λ}$ . For any Specht label $μ$ in $B$ , [F2] gives $C_{p} (μ) = λ$ and $∣ μ ∣ = ∣ λ ∣$ , so $μ = λ$ as well: the block has exactly one Specht row. Since $S_{k}^{λ}$ is indecomposable with head $D^{λ}$ by [F1] and lies in $B$ by hypothesis, the decomposition matrix of $B$ is the $1 \times 1$ matrix with entry $d_{λ λ} = [S_{k}^{λ} : D^{λ}] = 1$ , by the exact diagonal multiplicity in [F4]. A simple head alone would not establish multiplicity one.

2.1F1F5step 1.1algebra

The Cartan matrix and the projective cover. By [F5] applied to the block $B$ the Cartan matrix of $B$ is $C_{B} = D_{B}^{T} D_{B} = (1)$ , so the projective cover $P^{λ}$ of $D^{λ}$ has $D^{λ}$ as its only composition factor, with multiplicity one. A nonzero module of length one is simple, so $P^{λ} ≅ D^{λ}$ : the block $B$ has a projective simple module.

3.1F5F6step 1.1step 2.1

Defect zero and the matrix algebra. By step 2.1 the projective indecomposable $P^{λ}$ equals the simple $D^{λ}$ , and it is the unique projective indecomposable of $B$ because the simple modules of $B$ are in bijection with its projective indecomposables; so $D^{λ}$ is both the unique simple and the unique projective indecomposable of $B$ . The defect-zero criterion [F6] now gives that $B$ is a full matrix algebra over $k$ and that the defect group of $B$ is trivial, that is, $B$ has defect zero in the fixed splitting modular system.

4.1step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 give all the assertions: the single Specht label $λ$ , the decomposition matrix $(1)$ , the unique simple and projective indecomposable $D^{λ}$ , the Cartan matrix $(1)$ , the full matrix algebra structure and defect zero.

Depends on

Used by

Block weight is not the same as numerical defect Remark

Dependency tree · two levels

57 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

J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Mathematics 11, Chapters 1 and 4 (blocks, defect groups and the Cartan matrix) (standard reference, not scraped)
G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer 1978 (complete author-hosted scan, 161 pp.) (standard reference, not scraped)