斜平移四边形的中心问题，I
Central aspects of skew translation quadrangles, I
摘要：
除了埃尔米特建筑物$\mathcal{H}(4,q^2)$之外，直到对偶、平移对偶或佩恩积分的组合，每个已知的有限建筑物类型$\mathbb{B}_2$都满足一组一般综合性质，通常放在术语“斜平移广义四边形”（STGQ）中。在本系列论文中，我们对有限斜平移广义四边形进行了分类。在第一部分中，作为本文所建立的机制的推论，我们得到了令人惊讶的结果：任何奇数阶斜平移四边形$(s,s)$都是辛四边形;我们确定了所有具有不同的群的斜平移四边形（佩恩在一个不太一般的环境中提出的问题）;（c）发展了斜平移四边形根的结构理论，这也将在后面的部分中使用，它本质上告诉我们，一个非常一般的斜平移四边形类允许每个成员的理论最大根数，因此所有成员都是"中心"的。（控制STGQ所需的主要属性，如下文所示）;（d）我们解决了一类STGQ的主参数猜想，这类STGQ包含前一项的类，并且与所有STGQ的类在理论上一致。
英文摘要：
Except for the Hermitian buildings $\mathcal{H}(4,q^2)$, up to a combination of duality, translation duality or Payne integration, every known finite building of type $\mathbb{B}_2$ satisfies a set of general synthetic properties, usually put together in the term "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify finite skew translation generalized quadrangles. In the first installment of the series, as corollaries of the machinery we develop in the present paper, (a) we obtain the surprising result that any skew translation quadrangle of odd order $(s,s)$ is a symplectic quadrangle; (b) we determine all skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (c) we develop a structure theory for root-elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root-elations for each member, and hence all members are "central" (the main property needed to control STGQs, as which will be shown throughout); (d) we solve the Main Parameter Conjecture for a class of STGQs containing the class of the previous item, and which conjecturally coincides with the class of all STGQs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
