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Relative characters for H-projective RG-lattices

Published online by Cambridge University Press:  24 October 2008

Peter Symonds
Peter Symonds
Affiliation:
Ohio State University, Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210, U.S.A.
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Extract

If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 2 , September 1988 , pp. 207 - 213
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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