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Modular Specht form and radical quotient

Definition

Fix a prime $p$ , an integer $n \geq 0$ , and a splitting $p$ -modular system $(K, O, k)$ for $S_{n}$ , so that $K$ and $k$ are splitting fields for $S_{n}$ and its subgroups (A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras). Write

$M_{k}^{λ} : = k \otimes_{Z} M_{Z}^{λ}, S_{k}^{λ} : = k \otimes_{Z} S_{Z}^{λ} \subseteq M_{k}^{λ}$

for the base change of the integral tabloid module and Specht lattice; the inclusion is the injective polytabloid-span inclusion of Integral Specht lattice and base change, and $S_{k}^{λ}$ has the images of the standard polytabloids as $k$ -basis. Let $β_{k} : M_{k}^{λ} \times M_{k}^{λ} \to k$ be the reduced integral tabloid form, the unique $k$ -bilinear form with orthonormal tabloid basis (Integral tabloid form and Specht Gram matrix); it is symmetric, nondegenerate, and $S_{n}$ -invariant.

For a $k$ -subspace $V \subseteq M_{k}^{λ}$ put

$V^{⊥} : = {x \in M_{k}^{λ} : β_{k} (x, v) = 0 for all v \in V} .$

The form radical of $S_{k}^{λ}$ is

$R^{λ} : = S_{k}^{λ} \cap (S_{k}^{λ})^{⊥},$

and the modular Specht quotient (or James quotient) is

$D^{λ} : = S_{k}^{λ} / R^{λ} .$

The quotient is a $k [S_{n}]$ -module and may be zero. Its dimension is the $p$ -rank of the integral Gram matrix: if $G_{λ}$ is the Gram matrix of $β$ in the standard-polytabloid basis and $\overline{G_{λ}}$ its reduction modulo $p$ , i.e. the matrix of $β_{k}$ in the standard basis of $S_{k}^{λ}$ , then

$\dim_{k} D^{λ} = {rank}_{k} \overline{G_{λ}} = {rank}_{k} (G_{λ} m o d p) .$

Everything is defined by scalar extension: $S_{k}^{λ}$ and $β_{k}$ are determined by $λ$ and $k$ , and no choice of lifts of elements of $S_{k}^{λ}$ enters. $R^{λ}$ is a form radical, and is an $S_{n}$ -submodule; the identification of $R^{λ}$ with the module radical $rad (S_{k}^{λ}) = J (k [S_{n}]) S_{k}^{λ}$ (The radical, socle, head, and Loewy series of a finite-dimensional module) is a later consequence, asserted only when $D^{λ} \neq 0$ . In particular no simplicity of $D^{λ}$ is claimed here.

Facts & Assumptions

Given: A prime $p$ , an integer $n \geq 0$ , a partition $λ ⊢ n$ , a splitting $p$ -modular system $(K, O, k)$ for $S_{n}$ , and the definitions above.

[F1]

A splitting $p$ -modular system for $S_{n}$ has $k$ of characteristic $p$ and both $K$ and $k$ splitting fields for every subgroup of $S_{n}$ (A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras).

[F2]

For every commutative ring $R$ the natural map $R \otimes_{Z} S_{Z}^{λ} \to M_{R}^{λ}$ is injective onto the polytabloid span, with the images of the standard polytabloids as a basis; all elements are $R$ -linear combinations of standard polytabloids (Integral Specht lattice and base change).

[F3]

The reduced form $β_{k}$ has orthonormal tabloid basis, is symmetric, nondegenerate, $S_{n}$ -invariant, and its Gram matrix in the standard basis of $S_{k}^{λ}$ is the entrywise reduction of $G_{λ}$ (Integral tabloid form and Specht Gram matrix).

[F4]

The module radical of a finite-dimensional left $A$ -module is $rad (M) = J (A) M$ (The radical, socle, head, and Loewy series of a finite-dimensional module).

[F5]

Rank-nullity for a linear map with finite-dimensional domain gives $\dim V = rank T + \dim \ker T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F6]

The rank of a matrix equals the rank of the linear map it defines (The rank of a matrix equals the rank of the linear map $x \mapsto A x$ ).

Proof

technique · direct

1.1givenF1F2F3

The field $k$ is the residue field of the splitting $p$ -modular system fixed above, hence has characteristic $p$ and is a splitting field for $S_{n}$ and all its subgroups by [F1]. Thus $M_{k}^{λ}$ is a finite-dimensional $k$ -vector space with the tabloids as basis, and $S_{k}^{λ}$ is the $k$ -span of the polytabloids with the standard polytabloids as basis by [F2]. The form $β_{k}$ of [F3] is nondegenerate and $S_{n}$ -invariant.

1.2givenF3algebra

If $U \subseteq M_{k}^{λ}$ is an $S_{n}$ -submodule, then $U^{⊥}$ is an $S_{n}$ -submodule: for $x \in U^{⊥}$ , $u \in U$ and $σ \in S_{n}$ , invariance gives $β_{k} (σ x, u) = β_{k} (x, σ^{- 1} u) = 0$ because $σ^{- 1} u \in U$ , so $σ x \in U^{⊥}$ .

2.1givenF2step 1.1step 1.2

Since $S_{k}^{λ}$ is an $S_{n}$ -submodule by [F2] and [F3], step 1.2 shows that $(S_{k}^{λ})^{⊥}$ is an $S_{n}$ -submodule, hence so is $R^{λ} = S_{k}^{λ} \cap (S_{k}^{λ})^{⊥}$ ; the quotient $D^{λ} = S_{k}^{λ} / R^{λ}$ is therefore a $k [S_{n}]$ -module, and $D^{λ} = 0$ holds exactly when $S_{k}^{λ} \subseteq (S_{k}^{λ})^{⊥}$ .

2.2givenF3F5F6step 1.1

Let $φ : S_{k}^{λ} \to (S_{k}^{λ})^{*}$ be $φ (v) = β_{k} (v, \cdot) ∣_{S_{k}^{λ}}$ . Its kernel is exactly $R^{λ}$ , and in the standard basis of $S_{k}^{λ}$ paired with its dual basis the matrix of $φ$ is $\overline{G_{λ}}$ , so by [F6] the rank of $φ$ equals ${rank}_{k} \overline{G_{λ}}$ . Rank-nullity [F5] gives $\dim_{k} S_{k}^{λ} = {rank}_{k} \overline{G_{λ}} + \dim_{k} R^{λ}$ , hence $\dim_{k} D^{λ} = {rank}_{k} \overline{G_{λ}}$ .

3.1givenF2F3step 2.1step 2.2

The construction involves no choices of lifts: $M_{k}^{λ}$ , $S_{k}^{λ}$ and $β_{k}$ are obtained from the integral objects by scalar extension, and by [F2] every element of $S_{k}^{λ}$ is a $k$ -combination of the standard polytabloids, so the descriptions of $R^{λ}$ , $D^{λ}$ and $\dim_{k} D^{λ}$ depend only on $λ$ , $p$ and $k$ .

4.1givenF4step 2.1step 2.2∎

By step 2.1 the form radical is an $S_{n}$ -submodule and $D^{λ}$ is defined for every $λ$ , possibly zero; by step 2.2 its dimension is the rank of the reduced Gram matrix. The identification $R^{λ} = rad (S_{k}^{λ})$ with the radical of [F4] is asserted only later, for the nonzero case; nothing in this definition presupposes it.

Depends on

Used by

Modular Specht modules need not be simple, and form heads can vanish Counterexample
Decomposition matrices of S3 at p=2 and p=3 Example
Specht Gram rank for shape (2,2) Example
Nonzero maps into tabloid quotients force dominance Lemma
p-regular and p-restricted labels under transpose and sign Proposition
Dominance unitriangularity of the symmetric-group decomposition matrix Theorem
James submodule theorem over every field Theorem
Modular simple modules of the symmetric group Theorem
Nonzero modular Specht quotient criterion Theorem

Dependency tree · two levels

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Sources

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, §11.2 definition of D^lambda and §11.6 p-rank formula, printed pp. 39-40 (standard reference, not scraped)
Stacey Law, notes by Leonard Tomczak, Representation Theory of Symmetric Groups, §2.2 James submodule theorem and Gram-rank identity for S^lambda/(S^lambda\cap(S^lambda)^perp), printed pp. 13-14 (standard reference, not scraped)