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Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

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Abstract

Let F be a finite field of characteristic p and K a field which contains a primitive pth root of unity and char Kp. Suppose that a classical group G acts on the F-vector space V. Then it can induce the actions on the vector space V

V and on the group algebra K[V

V], respectively. In this paper we determine the structure of G-invariant ideals of the group algebra K[V

V], and establish the relationship between the invariant ideals of K[V] and the vector invariant ideals of K[V

V], if G is a unitary group or orthogonal group. Combining the results obtained by Nan and Zeng (2013), we solve the problem of vector invariant ideals for all classical groups over finite fields.

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Correspondence to Lingli Zeng.

Additional information

The research has been supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 16JK1789), and the National Natural Science Foundation of China (Program No. 11371343).

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Zeng, L., Nan, J. Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups. Czech Math J 66, 1059–1078 (2016). https://doi.org/10.1007/s10587-016-0310-x

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  • DOI: https://doi.org/10.1007/s10587-016-0310-x

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