Regulator constants of integral representations of finite groups
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 selfdual, called regulator constants. These were originally introduced by DokchitserDokchitser 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 nontrivial kernel. But, if G has cyclic Sylow psubgroups and we restrict to considering permutation lattices, then we show that the pairing is nondegenerate 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 selfdual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.
 Publication:

Mathematical Proceedings of the Cambridge Philosophical Society
 Pub Date:
 January 2020
 DOI:
 10.1017/S0305004118000579
 arXiv:
 arXiv:1703.10602
 Bibcode:
 2020MPCPS.168...75T
 Keywords:

 Mathematics  Representation Theory;
 20C10;
 19A22
 EPrint:
 43 pages. Restated the main theorem (Thm 6.8) in terms of $\mathbb{Z}_p$lattices as opposed to $\mathbb{Z}_{(p)}$lattices and added Section 6.3 providing criteria for the theorem to apply. To appear in Math. Proc. Cambridge Philos. Soc