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Virtual Weyl combinations detect q-Schur block separation

Statement

Let $F$ be a field, $q \in F^{\times}$ , $d, n \geq 1$ , and let $S = S (d, n)$ be the $q$ -Schur algebra over $F$ with Weyl modules $W^{λ}$ for $λ \in Λ^{+} (d, n)$ (The type-A q-Schur algebra and its Weyl modules). Let $M$ be a finite dimensional right $S$ -module all of whose composition factors lie in a single block $B$ of $S$ (Blocks of a finite-dimensional algebra via primitive central idempotents), and let integers $a_{λ}$ , $λ \in Λ^{+} (d, n)$ , satisfy $[M] = \sum_{λ \in Λ^{+} (d, n)} a_{λ} [W^{λ}]$ in the Grothendieck group $G_{0} (S -mod)$ (Grothendieck group of an essentially small abelian category). Then $W^{λ}$ lies in $B$ whenever $a_{λ} \neq 0$ . In particular no nonzero virtual combination of Weyl classes all of whose terms lie outside $B$ can equal the class of a module in $B$ .

No choice principle is used.

Facts & Assumptions

Given: A field $F$ , $q \in F^{\times}$ , $d, n \geq 1$ , the algebra $S = S (d, n)$ with Weyl modules $W^{λ}$ , $λ \in Λ^{+} (d, n)$ , a finite dimensional right $S$ -module $M$ whose composition factors all lie in one block $B$ , and integers $a_{λ}$ with $[M] = \sum_{λ} a_{λ} [W^{λ}]$ in $G_{0} (S -mod)$ (The type-A q-Schur algebra and its Weyl modules, Grothendieck group of an essentially small abelian category).

[F1]

Blocks: $1 = \sum_{i} ϵ_{i}$ is the unique decomposition of the identity of $S$ into pairwise orthogonal primitive central idempotents, the blocks are the two-sided ideals $B_{i} = S ϵ_{i} = ϵ_{i} S$ , and for every right $S$ -module $N$ one has $N = ⨁_{i} N ϵ_{i}$ ; a simple module $L$ lies in exactly one block and satisfies $L ϵ_{i} = L$ for that $i$ (Blocks of a finite-dimensional algebra via primitive central idempotents). We write $ϵ_{B}$ for the idempotent of the block $B$ of the statement.

[F2]

Each Weyl module has nonzero simple head $L^{λ}$ (The type-A q-Schur algebra and its Weyl modules).

[F3]

The proof of Semistandard weight dimensions detect virtual Weyl classes, steps 1.1 and 3.1, gives additive weight dimensions and the weight matrix $K_{λ ν} = \dim (W^{λ} 1_{ν})$ . This matrix, restricted to partitions with at most $d$ parts, is unitriangular and hence invertible by The Kostka change of basis is dominance-unitriangular. Applying these dimensions to a zero virtual Weyl combination forces every coefficient to vanish. This proves the linear independence used here; no Weyl filtration of an arbitrary module is assumed.

[F4]

An exact functor between abelian categories induces a homomorphism of Grothendieck groups, and the finite-dimensional $S$ -modules form an abelian category in which $G_{0}$ is computed by composition multiplicities (Grothendieck group of an essentially small abelian category).

Proof

technique · direct

1.1F1F4

Multiplication by the central idempotent $ϵ_{B}$ is the projection of every right $S$ -module onto its $B$ -summand: $N = ⨁_{i} N ϵ_{i}$ by [F1], so $N ϵ_{B}$ is that direct summand and is a submodule, and $N ϵ_{B} = N$ exactly when all composition factors of $N$ lie in $B$ . The assignment $N \mapsto N ϵ_{B}$ is exact, because it is the projection onto a direct summand of the identity functor: for $f : N \to N^{'}$ one has $f (N ϵ_{B}) = f (N) ϵ_{B}$ , $\ker (f ∣_{N ϵ_{B}}) = (\ker f) ϵ_{B}$ and $im (f ∣_{N ϵ_{B}}) = (im f) ϵ_{B}$ , so it preserves kernels and images.

1.2F1F2algebra

A nonzero finite-dimensional module has a simple quotient: choose a proper submodule of maximal vector-space dimension. If a Weyl module had two nonzero summands, each would have a nonzero head, so its head would not be simple, contrary to [F2]. Thus it is indecomposable. Its block decomposition [F1] has precisely one nonzero summand, the block containing its head $L^{λ}$ ; hence $W^{λ} ϵ_{B}$ is either $W^{λ}$ or $0$ .

2.1F4step 1.1step 1.2algebra

Apply the exact projection of step 1.1 to the class identity of the hypothesis. Since $M ϵ_{B} = M$ by the hypothesis that all composition factors of $M$ lie in $B$ , and since $W^{λ} ϵ_{B}$ is $W^{λ}$ or $0$ by step 1.2, the induced identity in $G_{0}$ reads $[M] = \sum_{λ : W^{λ} \in B} a_{λ} [W^{λ}] .$ Subtracting it from the given identity $[M] = \sum_{λ} a_{λ} [W^{λ}]$ yields $\sum_{λ : W^{λ} \notin B} a_{λ} [W^{λ}] = 0$ in $G_{0} (S -mod)$ .

3.1F3step 2.1∎

By the linear independence of the Weyl classes [F3], all coefficients in the vanishing combination of step 2.1 are zero: $a_{λ} = 0$ for every $λ$ with $W^{λ} \notin B$ , which is the contrapositive of the assertion that $a_{λ} \neq 0$ implies $W^{λ} \in B$ .

Depends on

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Sources

Andrew Mathas, Hecke algebras and Schur algebras of the symmetric group, complete 79-page prepublication manuscript (1998; expanded AMS University Lecture Series 15 edition, 1999) (standard reference, not scraped)
Sinead Lyle and Andrew Mathas, Blocks of affine and cyclotomic Hecke algebras, author-hosted complete 19-page preprint (published in Math. Z.) (standard reference, not scraped)