Abstract
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 1 G 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.
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Zhao, W., Willems, R. An analogue of the Duistermaat-van der Kallen theorem for group algebras. centr.eur.j.math. 10, 974–986 (2012). https://doi.org/10.2478/s11533-012-0028-4
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DOI: https://doi.org/10.2478/s11533-012-0028-4