简并能级
线性子空间
厄米矩阵
组合数学
子空间拓扑
辛几何
倍线性形式
领域(数学)
向量空间
数学
物理
有限域
空格(标点符号)
数学物理
纯数学
量子力学
数学分析
哲学
语言学
标识
DOI:10.1016/j.focat.2022.05.045
摘要
Let V be a d-dimensional vector space over a finite field F equipped with a non-degenerate hermitian, alternating, or quadratic form. Suppose |F|=q2 if V is hermitian, and |F|=q otherwise. Given integers e,e′ such that e+e′⩽d, we estimate the proportion of pairs (U,U′), where U is a non-degenerate e-subspace of V and U′ is a non-degenerate e′-subspace of V, such that U∩U′=0 and U⊕U′ is non-degenerate (the sum U⊕U′ is direct and usually not perpendicular). The proportion is shown to be positive and at least 1−c/q>0 for some constant c. For example, c=7/4 suffices in both the unitary and symplectic cases. The arguments in the orthogonal case are delicate and assume that dim(U) and dim(U′) are even, an assumption relevant for an algorithmic application (which we discuss) for recognising finite classical groups. We also describe how recognising a classical groups G relies on a connection between certain pairs (U,U′) of non-degenerate subspaces and certain pairs (g,g′)∈G2 of group elements where U=im(g−1) and U′=im(g′−1).
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