Abstract
Let F be a finite field of characteristic p and K a field which contains a primitive pth root of unity and char K ≠ p. 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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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