Defect 3 Blocks of Symmetric Group Algebras

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

“…These are largely concerned with comparing the decomposition numbers for blocks forming a [3 : κ]-pair. The results are largely the same as those in [13], but we are able to give quicker proofs using the modular branching rules.…”

“…In this section we define some notation for partitions in blocks of weight 3; this is similar to the notation used by Martin and Russell [13], but we use the numbering of runners described in §1. 1.6.…”

“…This result has been conjectured for some time and has proved elusive until now. In the special case of symmetric group algebras, Martin and Russell [13] have published a purported proof of this result; however, various errors have subsequently been found in that paper. In particular, when e = p = 5, λ = (8 2 , 4, 1) and µ = (12,9), the decomposition number [S λ : D µ ] is hard to calculate; it was eventually found to be 1 rather than 2 by a large computer calculation carried out by Lübeck and Müller.…”

Filtrations in Rouquier Blocks of Symmetric Groups and Schur Algebras

“…S. Martin [18] conjectured that the principal indecomposable modules in a weight w p-block of symmetric group algebra with w < p have a common radical length 2w + 1; this is clear for w = 0 and w = 1, and J. Scopes [24] proves the case of w = 2, while Martin and the second author [19,20] (rad ∆,ν,λ (v))…”

The ordinary quiver of a weight three block of the symmetric group is bipartite

“…Martin and Russell [15] initiated the study of blocks of weight 3 with the construction of the Ext 1 -quiver of the principal block of S 3p where p 5. They also attempted to show that the decomposition numbers for a weight 3 block of Abelian defect are at most 1 [16], but mistakes have been found in their proof. The complete proof was finally announced by the first author of the present paper in [9].…”