Conjugacy class properties of the extension of GL(n,q) generated by the inverse transpose involution
Conjugacy class properties of the extension of GL(n,q) generated by the inverse transpose involution
复制标题
转置对合生成的 GL(n,q) 扩展的共轭类性质
DOI：
10.1016/j.jalgebra.2003.07.004
复制
发表时间：
2003
影响因子：
0.9
通讯作者：
R. Guralnick
中科院分区：
文献类型：
--
作者：
Jason E. Fulman;R. Guralnick
令τ表示GL（n,q）的逆转置自同构，得到GL（n,q）中g的个数使ggτ等于给定元素h的公式。这推广了Gow和Macdonald在h为单位元的特殊情况下的结果。我们得出结论，对于g随机，ggτ表现为辛群和正交群的混合。结果表明，该公式适用于循环指数生成函数和渐近函数，并与随机划分理论有关。推导利用了GL（n,q）的表示理论模型和对称函数理论模型，包括Hall-Littlewood多项式的一个新的恒等式。在由逆转置自同构生成的GL（n,q）的扩展中，我们得到了偶特征有限辛群的随机元素的信息，以及共轭类数和中心化器大小的显式界。我们用域上双线性形式理论给出了这些结果的第二种方法。本文的结果是作者在即将开展的关于几乎简单群的行动的无序的工作的关键工具，我们在这个方向上给出了几个例子。
Letting τ denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that ggτis equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, ggτbehaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n,q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction.