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Normal-node composition multiplicities under residue restriction

Statement

Let $k$ be a field of characteristic $p > 0$ , $μ$ a $p$ -restricted partition of $d$ and $A$ a removable $i$ -node of $μ$ such that $ν = μ ∖ A$ is again $p$ -restricted; let $D (μ)$ be the simple $k S_{d}$ -module labelled by $μ$ in the dual-Specht (cellular) convention, so that $D (μ) ≅ D^{μ^{'}} \otimes sgn$ (p-regular and p-restricted labels under transpose and sign, Modular simple modules of the symmetric group), and let $E_{i}$ be the residue-summand functor of Generalized Jucys-Murphy primary idempotents and the residue-summand functors $E_{i}$ , $F_{i}$ . Then $[E_{i} D (μ) : D (ν)] = 0$ if $A$ is not a normal $i$ -node of $μ$ , and $[E_{i} D (μ) : D (ν)] = # {B : B normal i -node of μ weakly below A}$ (counting $A$ itself) if $A$ is normal. The corresponding Specht-to-restriction Hom criterion (Brundan 1998 Theorem 2.5, equivalently Kleshchev II Theorem 0.4) is a distinct statement and is not asserted here.

Facts & Assumptions

Given: A field $k$ of characteristic $p > 0$ , a $p$ -restricted partition $μ$ of $d$ , a removable $i$ -node $A = (r, μ_{r})$ with $ν = μ ∖ A$ $p$ -restricted, and the residue-summand functor $E_{i}$ .

[F1]

${Res}_{S_{d - 1}}^{S_{d}} M = ⨁_{j} E_{j} M$ , and $E_{j} M$ is the sum of the generalized eigenspaces of the last Jucys-Murphy element $L_{d} = T_{d} - T_{d - 1}$ with eigenvalue $j$ , for every finite-dimensional $k S_{d}$ -module $M$ ; here $T_{m} = \sum_{1 \leq a < b \leq m} (a b)$ is the transposition class sum of $S_{m}$ (Generalized Jucys-Murphy primary idempotents and the residue-summand functors $E_{i}$ , $F_{i}$ , The Jucys-Murphy elements of the symmetric group algebra).

[F2]

The central transposition class sum $T_{m}$ acts on the ordinary integral Specht lattice and every field specialization by the total content $z_{ξ} = \sum_{(r, c) \in [ξ]} (c - r)$ (The transposition class sum acts on a complex Specht module by total content). At $m = 0, 1$ , both the class sum and total content are zero directly. The actual same-shape perfect pairing identifies the Murphy cell $C_{M} (ξ)$ with the star dual of the ordinary Specht module (Murphy cells are the same-shape contragredient Specht duals). Since $T_{m}^{*} = T_{m}$ , the same scalar acts on that dual and its nonzero form quotient $D (ξ)$ . This uses the ordinary module itself, not a false ordinary/Murphy equality of form radicals.

[F3]

For a $p$ -restricted $ν$ , the module $D (ν)$ is nonzero, simple, self-dual and absolutely irreducible, so every $k S_{d - 1}$ -endomorphism of $D (ν)$ is a scalar (Modular simple modules of the symmetric group, p-regular and p-restricted labels under transpose and sign).

[F4]

The complete locally proved piecewise count is Kleshchev's first-level branching multiplicity. For the ordinary regular label $λ = μ^{'}$ and the transposed removable node $A^{'}$ , its multiplicity is zero off normality and counts the same-residue normal nodes weakly above $A^{'}$ on normality. The group sign/conjugate dictionary and signature order in [F5] transport this exact count; no bibliographic normal-node theorem is a proof premise.

[F5]

The dictionary between the two conventions: transposing exchanges removable nodes of $λ$ with removable nodes of $λ^{'}$ , exchanges "strictly above" for the dual order of the same residue class, and negates residues, $c - r \mapsto r - c$ ; ordinary normality on $λ$ in Kleshchev's first-level branching multiplicity is defined by signature normality of the transposed node on $λ^{'}$ . Brundan's lattice convention instead agrees with our signature on the same restricted shape, with residue names negated (Residue signatures, normal, good, conormal and cogood nodes, p-regular and p-restricted labels under transpose and sign).

Proof

technique · direct

1.1F1F2F3given

Localization of an occurring factor. On $D (μ)$ the class sum $T_{d}$ acts by $z_{μ}$ by [F2], hence it acts by that scalar on the whole restriction. The operator $L_{d} = T_{d} - T_{d - 1}$ on this restriction is therefore $z_{μ} - T_{d - 1}$ , a central $k S_{d - 1}$ operator preserving every submodule. If $D (ν)$ occurs as a composition factor, [F2] gives its $T_{d - 1}$ scalar $z_{ν}$ . Thus $L_{d}$ acts on this factor by $z_{μ} - z_{ν} = c (A) - r (A) = i$ . The generalized primary projections are exact by [F1]; every occurrence of this factor lies in $E_{i} D (μ)$ , and none in another summand. Consequently $[E_{i} D (μ) : D (ν)] = [D (μ) ↓_{S_{d - 1}} : D (ν)]$ , also when both sides are zero. This does not assert that a factor occurs before its multiplicity is computed.

2.1F4F5step 1.1algebra

Apply the local count [F4] to $λ = μ^{'}$ and the ordinary node $A^{'}$ . The group dictionary is $D (μ) ≅ D^{μ^{'}} \otimes sgn$ and likewise for $ν$ ; sign restricts to sign, so simultaneous twist preserves the restriction multiplicity. Transposing negates residues and reverses the order of corners in a fixed residue class, as [F5] states. The ordinary piecewise formula is therefore zero when $A$ is not normal, and otherwise is the number of normal $i$ -nodes weakly below $A$ , counting $A$ itself. Combine this with step 1.1. The unconditioned weakly-below count is not used off normality.

3.1F4F5step 2.1algebra∎

For $p = 3$ and $μ = (3, 3, 2, 1, 1)$ , the normal residue- $2$ nodes are $(3, 2)$ and $(5, 1)$ ; the same-residue signature has the two minus signs and no plus sign. The two respective regular removals give multiplicities $2$ and $1$ by step 2.1. At degree one the same argument gives the one-dimensional restriction to $S_{0}$ ; degree zero has no removable node. Thus the boundary cases and both displayed cases are included. No Specht-to-restriction Hom criterion is asserted.

Remarks

The complete count proof is the local first-level supplier, built from the integral root-interval basis and finite essential-subset recurrence. The cited Kleshchev and Brundan papers document the historical source of this formula; they are not external proof premises. The new local contribution here is the conditional class-sum localization and the exact transpose/sign transport. No unconditional count or implication from a zero form quotient to a zero actual module head is used.

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