Archive for December, 2001
Coxeter Transformations in Quantum Groups
by Xiao, Jie; Yang, Shilin
It is natural to define a Coxeter transformation in quantum groups by the product of Lusztig’s symmetries in some order. In this note, we show that the Ringel–Hall algebra approach enables us to determine the behavior of its action in case of finite type.
DOI: 10.1023/A:1012753020062
Print publication date: 12/1/2001
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Character Relations and Simple Modules in the Auslander–Reiten Graph of the Symmetric and Alternating Groups and their Covering Groups
by Bessenrodt, C.; Uno, K.
From character relations for symmetric groups or Hecke algebras such as the Murnaghan–Nakayama formula and the Jantzen–Schaper formula, we obtain a lower bound for the diagonal entries of Cartan matrices. Moreover, we prove an analogous character relation for covering groups of symmetric groups and obtain a similar lower bound. As an application, we show in these situations that for wild blocks simple modules must lie at the end of the Auslander–Reiten quiver, which is equivalent to the fact that the hearts of projective indecomposable modules are indecomposable.
DOI: 10.1023/A:1012701002315
Print publication date: 12/1/2001
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Derived Lengths of Solvable Groups Having Five Irreducible Character Degrees I
by Lewis, Mark L.
Let G be a solvable group with five character degrees. We show that the derived length of G is at most 5. This verifies that the Taketa inequality, dl(G)≤|cd(G)|, is valid for solvable groups with at most five character degrees.
DOI: 10.1023/A:1012706718244
Print publication date: 12/1/2001
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Relative Projectivity of Modules and Cohomology Theory of Finite Groups
by Okuyama, Tetsuro; Sasaki, Hiroki
In the modular representation theory of finite groups the theory of projectivity relative to subgroups is of fundamental importance. To generalize this notion the theory of projectivity relative to “modules” was introduced by the first author. Our aim is to show some aspects of cohomology theory of finite groups concerning projectivity of modules relative to both subgroups and modules. We shall give some applications to the cohomology theory; especially we shall calculate the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups.
DOI: 10.1023/A:1012746505198
Print publication date: 12/1/2001
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Contents of Volume 4 (2001)
by
DOI: 10.1023/A:1017310708892
Print publication date: 12/1/2001
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