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# Hecke algebras and class-group invariant

Published:1997-12-01
Printed: Dec 1997
• V. P. Snaith
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## Abstract

Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide.
 MSC Classifications: 16S34 - Group rings [See also 20C05, 20C07], Laurent polynomial rings 19A99 - None of the above, but in this section 11R65 - Class groups and Picard groups of orders