An analogue of the Duistermaat-van der Kallen theorem for group algebras

作者：Zhao Wenhua^*; Willems Roel

来源：Central European Journal of Mathematics, 2012, 10(3): 974-986.

DOI：10.2478/s11533-012-0028-4

• The Duistermaat-van der Kallen Theorem
• Mathieu subspaces
• Groups algebras

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: Algebre non Commutative, Groupes Quantiques et Invariants, Reims, June 26-30, 1995, Semin. Congr., 2, Societe Mathematique 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 = Z(n), n %26gt;= 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.

• 出版日期2012-6