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Hecke Specht theory specialises to group Specht theory at q = 1

Statement

Let $F$ be a field, $q = 1 \in F$ , and let $H_{F, 1} (S_{n})$ be the specialization of the generic type-A Hecke algebra at $v \mapsto 1$ , so that $H_{F, 1} (S_{n}) ≅ F [S_{n}]$ with $T_{w} \mapsto w$ (Polytabloid specialization and the exact signed multiplicity dictionary, step 1.1). Then the Hecke theory of this page specializes as follows.

The Murphy elements $m_{μ} = \sum_{w \in S_{μ}} T_{w}$ specialize to the row symmetrizers $\sum_{w \in S_{μ}} w$ , and the Murphy basis ${m_{s t}}$ of $H_{F, 1} (S_{n})$ specializes to the Murphy basis of the group algebra $F [S_{n}]$ (the base change $Z [S_{n}] \to F [S_{n}]$ of the integral Murphy basis).
For every $λ ⊢ n$ the specialization of the page's Hecke Specht module $S^{λ}$ (The type-A Hecke Specht module and its cell form) is the ordinary Specht module in the Dipper-James convention: $S_{q = 1}^{λ} ≅ S_{D J}^{λ}$ as $F [S_{n}]$ -modules, with the standard polytabloid basis (Standard polytabloids form a basis of a complex Specht module); the specialized cell form is the standard bilinear form on $S^{λ}$ whose radical defines the James quotient $D^{λ}$ (Modular Specht form and radical quotient).
The quantum characteristic of $q = 1$ is $e = char (F)$ when $char (F) > 0$ , and $e = \infty$ when $char (F) = 0$ (The quantum characteristic of an invertible Hecke parameter); in characteristic $p > 0$ a partition is $e$ -regular exactly when it is $p$ -regular. Hence $D_{q = 1}^{λ} \neq 0$ exactly for $p$ -regular $λ$ (Nonzero modular Specht quotient criterion), and for non- $p$ -regular $λ$ one has $r a d S^{λ} = S^{λ}$ . Over splitting residue fields this also matches the group-side classification and conjugate-sign dictionary of Modular simple modules of the symmetric group and p-regular and p-restricted labels under transpose and sign; the assertion here holds over every field by the cellular classification below. In characteristic $0$ the specialization is split semisimple and all Specht modules are irreducible.

Facts & Assumptions

Given: A field $F$ and the specialization $H_{F, 1} (S_{n}) ≅ F [S_{n}]$ (Polytabloid specialization and the exact signed multiplicity dictionary, step 1.1).

[F1]

(Ordinary Specht modules are the signed conjugate Murphy cells, Polytabloid specialization and the exact signed multiplicity dictionary) The Dipper-James ordinary Specht module with its polytabloid basis and standard form is identified with the signed conjugate Murphy cell module of The type-A Hecke Specht module and its cell form by a base-change compatible isomorphism whose Laurent unit form factor is $q^{ℓ (w_{0} (S_{λ^{'}})) - ℓ (w_{λ})}$ , equal to $1$ at $q = 1$ over every field; hence at $q = 1$ the ordinary module and the standard form agree on the specialization, including on zero-form quotients.

[F2]

Over every field, the integral-polytabloid specialization and standard tabloid form are supplied by Polytabloid specialization and the exact signed multiplicity dictionary, clause 1 and proof step 2.1. Over a splitting residue field the form quotient is exactly Modular Specht form and radical quotient; Standard polytabloids form a basis of a complex Specht module records the complex-field basis. These latter items are used only within their stated coefficient fields.

[F3]

(The quantum characteristic of an invertible Hecke parameter) At $q = 1$ the quantum characteristic is $e = p = char (F)$ when $p > 0$ and $e = \infty$ when $p = 0$ ; a partition is $e$ -regular exactly when it is $p$ -regular.

[F4]

At $q = 1$ in positive characteristic, the ordinary group Specht radical quotient is nonzero exactly for $p$ -regular labels (Nonzero modular Specht quotient criterion). In characteristic zero $e = \infty$ , and the complete local separated-range proof gives nonsingular Murphy cell forms and split semisimplicity (Separated Hecke residues give seminormal idempotents and bases); the signed-conjugate form identity in [F1] transfers their nondegeneracy to every ordinary Specht module. No finite- $e$ Hecke nonzero-label criterion is used.

[F5]

Over every field, the nonzero cell-form quotients of a cellular algebra form a complete set of pairwise distinct absolutely simple modules (Nonzero cell heads classify the simple modules of a cellular algebra). The exact signed-conjugate equivalence of [F1] transports this classification to ordinary Specht quotients; no splitting-field-only group theorem is invoked over an arbitrary $F$ .

Proof

technique · direct

1.1givenF1algebra

Murphy basis: base change along $Z [v^{\pm 1}] \to F$ , $v \mapsto 1$ , sends $T_{w}$ to $w$ and $m_{μ} = \sum_{w \in S_{μ}} T_{w}$ to the row symmetrizer $\sum_{w \in S_{μ}} w$ ; the structure constants of the Murphy basis are polynomials in $q$ , so they specialize to structure constants in $F [S_{n}]$ and the specialized family ${m_{s t}}$ satisfies the cellular axioms over $F$ ; it is a basis because base change of the free integral Murphy basis is a basis, by The Murphy tableau elements form a cellular basis. This is part 1.

2.1step 1.1F1F2

Module and form: by [F1] the specialization of the page's Specht module is the ordinary Dipper-James Specht module with the standard polytabloid basis, and the specialized cell form differs from the standard form by the Laurent factor $q^{ℓ (w_{0} (S_{λ^{'}})) - ℓ (w_{λ})}$ , which is $1$ at $q = 1$ ; hence the two forms agree over every field and have the same radical, so the specialized quotient is the James quotient $D^{λ}$ . This is part 2.

3.1step 2.1F1F3F4F5∎

Labels: by [F3], $e = p$ in positive characteristic, and $e$ -regular means $p$ -regular. Step 2.1 identifies the specialized module and its form exactly with the ordinary group module and form, so the ordinary criterion in [F4] gives precisely the nonzero $p$ -regular quotients and the zero form on all other labels, These are all the simples by the cellular classification [F5] and the exact signed-conjugate equivalence of [F1]. In characteristic zero, $e = \infty > n$ , so the local separated-range proof in [F4] makes the Murphy cell forms nonsingular and the algebra split semisimple. The exact signed-conjugate unit form dictionary [F1] gives the same conclusion for all ordinary Specht modules. Thus all claims follow over the original field, without using the held finite- $e$ Hecke classifier.

Depends on

Used by

The Murphy cellular basis of H(S₃) Example
The Schur functor compares first-level branching with Hecke restriction Proposition
Nakayama block classification for symmetric groups Theorem

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