A fixed-point formula for the classical groups over a finite field
A fixed-point formula for the classical groups over a finite field
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有限域上经典群的定点公式
DOI：
10.1090/s0002-9947-1965-0172935-2
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发表时间：
1965
期刊：
影响因子：
--
通讯作者：
Louis Solomon
中科院分区：
文献类型：
--
作者：
Louis Solomon
其中 t 是不定数。在本文中，我们考虑有限域上酉群、辛群和正交群的此类公式的可能存在性。这个问题是很自然的，因为我们对特征 p > 0 的反射群几乎一无所知，并且由倍半线性形式定义的经典群都是由固定超平面的元素生成的。粗略地说，结果是，当且仅当形式的维特指数为 0 或 1 时，此类公式才存在。尽管酉、辛和正交情况的陈述和证明有相似之处，但尝试将它们放在一起处理是很尴尬的。在下面的定理中，我们认为 n 是固定的，并让 q 在素数幂集上变化。
where t is an indeterminate. In this paper we consider the possible existence of such formulas for the unitary, symplectic, and orthogonal groups over a finite field. The question is a natural one since we know next to nothing about reflection groups in characteristic p > 0, and the classical groups defined by a sesquilinear form are all generated by elements which fix a hyperplane. The results, roughly stated, are that such formulas exist if and only if the Witt index of the form is 0 or 1. Although there are similarities in the statements and proofs for the unitary, symplectic and orthogonal cases, it is awkward to try to handle them together. In the following theorems we consider n to be fixed and let q vary over the set of prime powers.