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Regulator constants of integral representations of finite groups

Published online by Cambridge University Press:  05 September 2018

ALEX TORZEWSKI
ALEX TORZEWSKI*
Affiliation:
Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL. e-mail: alex.torzewski@gmail.com
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Abstract

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 75 - 117
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Supported by a PhD studentship from the Engineering and Physical Sciences Research Council.

References

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