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The type-A Hecke decomposition matrix is dominance unitriangular

Statement

Let $F$ be a field, $q \in F^{\times}$ , $e = e (F, q)$ and $H = H_{F, q} (S_{n})$ (The quantum characteristic of an invertible Hecke parameter). For $λ ⊢ n$ and $e$ -regular $μ ⊢ n$ let $d_{λ μ} : = [S^{λ} : D^{μ}],$ the multiplicity of $D^{μ}$ in a composition series of the Specht module $S^{λ}$ (The type-A Hecke Specht module and its cell form, The simple type-A Hecke modules are indexed by e-regular partitions). Then $d_{μ μ} = 1 and d_{λ μ} \neq 0 ⟹ μ ⊵ λ,$ where $⊵$ is the dominance order on partitions (Dominance order on partitions): a simple factor $D^{μ}$ of $S^{λ}$ can occur only if the $e$ -regular label $μ$ dominates $λ$ . Equivalently the decomposition matrix $(d_{λ μ})$ is unitriangular along dominance. This constrains but does not compute the entries: no off-diagonal decomposition number is determined by cellularity.

Facts & Assumptions

Given: A field $F$ , $q \in F^{\times}$ , $H = H_{F, q} (S_{n})$ with Specht modules $S^{λ}$ and simple quotients $D^{μ}$ in the signed conjugate normalization (The type-A Hecke Specht module and its cell form), together with the dominance order on partitions of $n$ (Dominance order on partitions).

[F1]

(The type-A Hecke Specht module and its cell form) $S^{λ} = (S_{M}^{λ^{'}})^{θ}$ and $D^{μ} = (D_{M}^{μ^{'}})^{θ}$ for the conjugate partitions, where $S_{M}^{ν}$ is the Murphy cell module and $θ$ is the involutive automorphism $θ (T_{i}) = - q T_{i}^{- 1}$ ; pullback along $θ$ is exact and preserves submodules, quotients, composition multiplicities and simplicity.

[F2]

(Cellular decomposition matrices are unitriangular for the cell order) For the Murphy cellular basis of $H$ the composition multiplicity $[S_{M}^{ν} : D_{M}^{ρ}]$ satisfies $[S_{M}^{ρ} : D_{M}^{ρ}] = 1$ whenever $D_{M}^{ρ} \neq 0$ and is zero unless $ν ⊵ ρ$ .

[F3]

(Dominance order on partitions) Conjugation reverses dominance: $λ^{'} ⊵ μ^{'}$ if and only if $μ ⊵ λ$ .

[F4]

(The simple type-A Hecke modules are indexed by e-regular partitions) $D^{μ} \neq 0$ exactly for the $e$ -regular $μ$ .

Proof

technique · direct

1.1givenF1F4

Invariance of composition multiplicities under the normalisation: by [F1] the twist $(-)^{θ}$ is an exact equivalence of module categories with inverse $(-)^{θ}$ itself, so it is additive on short exact sequences and carries composition series to composition series; hence $[S^{λ} : D^{μ}] = [S_{M}^{λ^{'}} : D_{M}^{μ^{'}}]$ for all $λ ⊢ n$ and $e$ -regular $μ ⊢ n$ .

2.1step 1.1F2F3

Vanishing: if $d_{λ μ} \neq 0$ , then $[S_{M}^{λ^{'}} : D_{M}^{μ^{'}}] \neq 0$ by step 1.1, so $λ^{'} ⊵ μ^{'}$ by [F2], and conjugation reversal [F3] gives $μ ⊵ λ$ .

2.2step 1.1F2

Diagonal: taking $λ = μ$ in step 1.1 gives $d_{μ μ} = [S_{M}^{μ^{'}} : D_{M}^{μ^{'}}] = 1$ by [F2].

3.1F1F4step 2.1step 2.2∎

Assembly: steps 2.1 and 2.2 are exactly the two claims; ordering the rows and columns of $(d_{λ μ})$ by any linear extension of dominance displays the unitriangular shape, and no off-diagonal entry is determined by the cellular structure. At $q = 1$ , [F1] and [F4] identify the same ordinary group labels; over a splitting p-modular system this agrees with Dominance unitriangularity of the symmetric-group decomposition matrix within that supplier's hypotheses. The present proof establishes the bound over every field.

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