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The socle of an uncoloured modular restriction is the multiplicity-free sum over the good residues

Statement

Let $k$ be a field of characteristic $p > 0$ and let $μ$ be a $p$ -restricted partition of $d \geq 1$ ; write $D (μ)$ for the simple $k S_{d}$ -module labelled by $μ$ (Modular simple modules of the symmetric group, p-regular and p-restricted labels under transpose and sign). Let $A_{i}$ denote the good $i$ -node of $μ$ , for those $i \in Z / p$ with $ε_{i} (μ) > 0$ (Residue signatures, normal, good, conormal and cogood nodes). Then ${soc}_{S_{d - 1}} (D (μ) ↓_{S_{d - 1}}) = ⨁_{i : ε_{i} (μ) > 0} D (μ ∖ A_{i}),$ a multiplicity-free sum over distinct simple modules: distinct residues give distinct summands. The sum runs over the residues $i$ with $ε_{i} (μ) > 0$ , one good node per such residue; the completely reducible special case (the restriction is semisimple if and only if every normal node is good, Kleshchev II Theorem 0.6) is a separate statement and is not asserted here.

Facts & Assumptions

Given: A field $k$ of characteristic $p > 0$ , a $p$ -restricted partition $μ$ of $d \geq 1$ , and the good $i$ -nodes $A_{i}$ of $μ$ .

[F1]

${Res}_{S_{d - 1}}^{S_{d}} M = ⨁_{i \in Z / p} E_{i} M$ for every finite-dimensional $k S_{d}$ -module $M$ , where each $E_{i} M$ is a $k S_{d - 1}$ -submodule and $E_{i}$ is the residue-summand functor attached to the primary Jucys-Murphy idempotents (Generalized Jucys-Murphy primary idempotents and the residue-summand functors $E_{i}$ , $F_{i}$ ).

[F2]

For a residue $i$ with $ε_{i} (μ) > 0$ the good $i$ -node $A_{i}$ is the unique normal $i$ -node with no normal $i$ -node above it, and the nodes $A_{i}$ for distinct residues $i$ are distinct nodes of $μ$ , so the partitions $μ ∖ A_{i}$ of $d - 1$ are distinct (Residue signatures, normal, good, conormal and cogood nodes).

[F3]

Steps 1.1–17.1 of The cogood node gives the simple socle and head of the residue induction prove the restriction endpoint independently of this corollary: the root interval primitive lines, compensated containment, Schur-corner socle lift and class-sum localization give ${soc}_{S_{d - 1}} (E_{i} D (μ)) = {\begin{cases} D (μ ∖ A_{i}), & ε_{i} (μ) > 0, \\ 0, & ε_{i} (μ) = 0. \end{cases}$ The good-deletion labels are $p$ -restricted by that proof's step 11.1. Only this proved restriction argument is used here; the later induction endpoint and its conormal multiplicity are not needed.

[F4]

Over every field $k$ of characteristic $p$ , the conjugate-sign isomorphism of Conjugate Specht modules are sign-twisted duals over every field identifies $S (ν) = (S_{k}^{ν})^{*}$ with $S_{k}^{ν^{'}} \otimes sgn$ . Twisting by sign preserves submodules and their radicals, since it multiplies each group operator by a unit scalar. Thus its head is $D (ν) = D_{k}^{ν^{'}} \otimes sgn$ for $p$ -restricted $ν$ . These heads are nonzero and absolutely simple by Nonzero modular Specht quotient criterion. They are pairwise nonisomorphic: twisting an isomorphism back by sign and applying the separation assertion of Nonzero maps into tabloid quotients force dominance forces $ν_{1}^{'} = ν_{2}^{'}$ , hence $ν_{1} = ν_{2}$ . No splitting-field descent is assumed.

[F5]

The socle of a finite-dimensional module is the sum of its simple submodules, and the socle of a finite direct sum is the direct sum of the socles of the summands. Indeed each projection of a simple submodule is zero or simple, so that submodule lies in the sum of the summand socles; the reverse inclusion holds because each simple submodule of a summand is a submodule of the sum (The radical, socle, head, and Loewy series of a finite-dimensional module).

Proof

technique · direct

1.1givenF1F5

By [F1] the restriction decomposes as $Res D (μ) = ⨁_{i \in Z / p} E_{i} D (μ)$ with each $E_{i} D (μ)$ a $k S_{d - 1}$ -submodule, and by [F5] the socle of the restriction is the direct sum $⨁_{i} soc (E_{i} D (μ))$ .

1.2givenF3F4

By the local restriction argument [F3], for each residue $i$ the module $soc (E_{i} D (μ))$ is $D (μ ∖ A_{i})$ when $ε_{i} (μ) > 0$ and is $0$ when $ε_{i} (μ) = 0$ ; by [F3] each $μ ∖ A_{i}$ is a $p$ -restricted partition of $d - 1$ , so each displayed $D (μ ∖ A_{i})$ is a simple $k S_{d - 1}$ -module by [F4].

2.1step 1.1step 1.2F3

Summing the descriptions of step 1.2 over the finitely many residues $i \in Z / p$ and using step 1.1 gives ${soc}_{S_{d - 1}} (D (μ) ↓_{S_{d - 1}}) = ⨁_{i : ε_{i} (μ) > 0} D (μ ∖ A_{i}) .$

3.1step 2.1F2F4

The sum is multiplicity-free and its summands are pairwise non-isomorphic: if $i \neq j$ and both $ε_{i} (μ)$ and $ε_{j} (μ)$ are positive, then the good nodes $A_{i}$ and $A_{j}$ have different residues and are therefore distinct nodes of $μ$ , so the $p$ -restricted partitions $μ ∖ A_{i}$ and $μ ∖ A_{j}$ of $d - 1$ are distinct by [F2]; by [F4] their simple modules $D (μ ∖ A_{i})$ and $D (μ ∖ A_{j})$ are non-isomorphic. Each residue contributes its summand once, so each isomorphism class occurs with multiplicity one in the displayed socle.

4.1step 2.1step 3.1given∎

The displayed decomposition of step 2.1 has the left-hand side ${soc}_{S_{d - 1}} (D (μ) ↓_{S_{d - 1}})$ and, by step 3.1, is a multiplicity-free sum over distinct simple modules with one summand per residue $i$ satisfying $ε_{i} (μ) > 0$ . No claim is made here about the restriction being semisimple. In the smallest case $d = 1$ one has $μ = (1)$ , and the only node of residue $0$ is the removable node $(1, 1)$ , so $ε_{0} (μ) = 1$ and the right-hand side is the single term $D (\emptyset) = k$ , while the restriction to the trivial group $S_{0}$ is $k$ itself, in agreement with the display; this is the smallest instance in which the summands carry the empty partition as a label.

Remarks

Proof input. The per-residue endpoint is proved in steps 1.1–17.1 of The cogood node gives the simple socle and head of the residue induction. That argument does not depend on this corollary or on the later thm-good-node-gives-the-simple-socle-and-head-of-residue-restriction; the latter uses this corollary together with the normal-node count. The cited literature documents the origin of the result.
No character theory. Nothing in the proof uses complex characters or reciprocity; modular multiplicities are not computed from characteristic zero data.
Choice. All index sets are finite and no choice principle is used.

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