Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen theorem [Duistermaat J.J., van der Kallen W., Constant terms in powers of a Laurent polynomial, Indag. Math., 1998, 9(2), 221–231], and also by recent studies on the notion of Mathieu subspaces, we show that for finite groups G, V G also forms a Mathieu subspace of the group algebra R[G] when certain conditions on the base ring R are met. We also show that for the free abelian groups G = ℤn, n ≥ 1, and any integral domain R of positive characteristic, V G fails to be a Mathieu subspace of R[G], which is equivalent to saying that the Duistermaat-van der Kallen theorem cannot be generalized to any field or integral domain of positive characteristic.
@article{bwmeta1.element.doi-10_2478_s11533-012-0028-4,
author = {Wenhua Zhao and Roel Willems},
title = {An analogue of the Duistermaat-van der Kallen theorem for group algebras},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {974-986},
zbl = {1255.16022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0028-4}
}
Wenhua Zhao; Roel Willems. An analogue of the Duistermaat-van der Kallen theorem for group algebras. Open Mathematics, Tome 10 (2012) pp. 974-986. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0028-4/
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