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The Quarterly Journal of Mathematics
Article . 1995 . Peer-reviewed
Data sources: Crossref
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SYMMETRIC GROUP BLOCKS OF DEFECT TWO

Symmetric group blocks of defect two
Authors: Scopes, Joanna;

SYMMETRIC GROUP BLOCKS OF DEFECT TWO

Abstract

Using the techniques of the author's paper on Morita equivalence of blocks of symmetric groups the following strong results are proved for blocks of defect 2 in symmetric groups (corresponding results hold for the alternating groups). Theorem I. A block of a symmetric group with defect two in characteristic \(p > 0\) has the following properties. 1. All the decomposition numbers are 0 or 1. 2. A row in the decomposition matrix has at most 5 non-zero entries. 3. All diagonal entries in the Cartan matrix are greater than or equal to 3. 4. All non-diagonal entries in the Cartan matrix are less than or equal to 2. 5. The dimension of \(\text{Ext}^1 [D^\lambda, D^\mu]\) is 0 or 1 for all \(p\)-regular \(\lambda\) and \(\mu\). 6. The dimension of \(\text{Ext}^1 [D^\lambda, D^\mu]\) is 0 if \(D^\mu\) is isomorphic to \(D^\lambda\) or \(D^\lambda \otimes \text{alt}\). 7. All the projective indecomposable modules have Loewy length 5 and are stable, that is their Loewy series are the same as their socle series. Theorem II. Given any symmetric group block \(B\) of defect two there is an algorithm which one may apply to determine 1. the decomposition matrix of \(B\); 2. the Cartan matrix of \(B\); 3. the \(\text{Ext}^1\) quiver of \(B\); 4. the Loewy structures of the projective indecomposable modules of \(B\).

Keywords

Loewy series, Modular representations and characters, Representations of finite symmetric groups, Loewy length, blocks of defect 2, projective indecomposable modules, decomposition numbers, Cartan matrix, blocks of symmetric groups, alternating groups, decomposition matrix

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
28
Average
Top 10%
Average
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