On Certain Blocks of Schur Algebras

Abstract

In this paper, what is already known about defect 2 blocks of symmetric groups is used to deduce information about the corresponding blocks of Schur algebras. This information includes Ext-quivers and decomposition numbers, as well as Loewy structures of the Weyl modules, principal indecomposable modules and tilting modules.

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

“…Suppose λ and μ are p-regular partitions lying in a weight 3 block of S n , and that C is a block of S n−1 of weight less than 3. (2) .…”

“…By developing the combinatorics of these blocks and studying the application of the Jantzen-Schaper formula, he was able to give a simple description of the decomposition numbers. Chuang and the second author [2] used this to give a description of the Ext 1 -spaces between simple modules. The important consequence of this result for us is the following.…”

“…, μ (t) are totally ordered by the Jantzen-Schaper order, and we suppose that μ (1) · · · μ (t) , so that μ (1) = μ. If w is the weight of C, then we have t = 4 − w 2, and we shall be interested in the partition μ (2) . The next lemma relates the parities of μ, μ − and μ (2) .…”

“…If w is the weight of C, then we have t = 4 − w 2, and we shall be interested in the partition μ (2) . The next lemma relates the parities of μ, μ − and μ (2) . Lemma 2.21.…”

“…Blocks of weight 2 were studied by several authors, including Richards, who gave a combinatorial description of their decomposition numbers over a field of odd characteristic [21]. Chuang and the second author built on this work and were able to describe the quiver of an arbitrary weight 2 block in odd characteristic [2], verifying that all such quivers are 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.…”