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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modular permutation representations

Author: L. L. Scott
Journal: Trans. Amer. Math. Soc. 175 (1973), 101-121
MSC: Primary 20C20
MathSciNet review: 0310051
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Abstract: A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras.

Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15].

A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauer's theory of blocks to Green's work on indecomposable modules.

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Keywords: Indecomposable module, vertex, centralizer ring, decomposition numbers, Brauer homomorphism, Brauer's first fundamental theorem, Green correspondence, Brauer's seconds fundamental theorem, defect group, defect 0 and 1, splitting field
Article copyright: © Copyright 1973 American Mathematical Society