Some Canonical Basis Vectors in the Basic Uq([formula]n)-Module

Decomposition numbers for weight three blocks of symmetric groups and Iwahori--Hecke algebras

“…The decomposition numbers for Rouquier blocks (of any weight) are known over a field of infinite characteristic [2,12]. In addition, a recent paper of James, Lyle and Mathas [8] shows that James's Conjecture holds for Rouquier blocks.…”

“…(1) Suppose λ is an e-regular partition lying in a block B, and that we can find (2) and Lemma 4.4. Our strategy will be to use induction on n (in the usual direction), but we shall also make use of the fact [8,Corollary 4] that the adjustment matrix for a Rouquier block is trivial.…”

Filtrations in Rouquier Blocks of Symmetric Groups and Schur Algebras

“…An independent proof in the case that µ is p-restricted is given in [3]. Because we have shown that the radical filtrations and Jantzen filtrations coincide (Proposition 5.5) we deduce that for λ, µ ∈ Λ(ρ, w),…”

“…Part (2) follows immediately from the Theorem 6.2(2). For part (3), define the parity of L(λ) to be the parity of ∑ j odd |λ j |. By part (1), if L(λ) extends L(µ), then they have different parities.…”

An Extension of James's Conjecture

“…By applying Theorem 2.14 e − 1 − i times and then using Theorem 2.11 (or by using the known decomposition numbers for Rouquier blocks in infinite characteristic [3,21]), we find that d e λµ = 0; by Proposition 2.10 (or by using the well-known decomposition numbers for blocks of weight 1), we get d ep λµ > 0. Lemma 2.7(1) now implies that there is some ξ in B with µ ξ λ such that the (ξ, µ)-entry of the adjustment matrix for B is non-zero.…”