Defect 3 blocks of symmetric group algebras. I. (English) Zbl 1024.20012
Consider the symmetric group algebra \(k{\mathfrak S}_n\) over an algebraically closed field \(k\) of non-zero characteristic \(p\). The first author [in Q. J. Math., Oxf. II. Ser. 44, No. 173, 87-99 (1993; Zbl 0835.20020)], conjectured that for a defect \(d\) block of \(k{\mathfrak S}_n\) (with \(p>d\)) its principal indecomposable modules have a common Loewy length of \(2d+1\). It is well-known that the conjecture holds for \(d=1\) and J. Scopes [Q. J. Math., Oxf. II. Ser. 46, No. 182, 201-234 (1995; Zbl 0835.20022)] verified it for \(d=2\). Here, expanding upon work of the first author and L. Russell [J. Algebra 185, No. 2, 440-480 (1996; Zbl 0894.20016)], the authors investigate the defect 3 case.
The authors first consider the principal block of \(k{\mathfrak S}_{3p}\) where they show that the Ext-quiver is bipartite and the principal indecomposable modules have common Loewy length 7. They then consider the more general case of a defect 3 block \(B\) of \(k{\mathfrak S}_n\) and a defect 3 block \(\widetilde B\) of \(k{\mathfrak S}_{n-1}\) that form a \([3:1]\)-pair and show that if the bipartite and Loewy length 7 properties hold for \(\widetilde B\) then they also do for \(B\). Using this result, they show inductively that the principal block of \(k{\mathfrak S}_n\) satisfies these properties for \(3p<n<4p\).
The authors first consider the principal block of \(k{\mathfrak S}_{3p}\) where they show that the Ext-quiver is bipartite and the principal indecomposable modules have common Loewy length 7. They then consider the more general case of a defect 3 block \(B\) of \(k{\mathfrak S}_n\) and a defect 3 block \(\widetilde B\) of \(k{\mathfrak S}_{n-1}\) that form a \([3:1]\)-pair and show that if the bipartite and Loewy length 7 properties hold for \(\widetilde B\) then they also do for \(B\). Using this result, they show inductively that the principal block of \(k{\mathfrak S}_n\) satisfies these properties for \(3p<n<4p\).
Reviewer: Christopher P.Bendel (Menomonie)
MSC:
| 20C30 | Representations of finite symmetric groups |
| 20C20 | Modular representations and characters |
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 20J06 | Cohomology of groups |
Keywords:
symmetric groups; Ext-quivers; group algebras; principal blocks; principal indecomposable modules; Loewy lengthsReferences:
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