二次型复合中Hurwitz问题与Yuzvinsky猜想的研究

本项目拟研究二次型复合中Hurwitz问题、Yuzvinsky猜想等相关问题，这类问题与众多数学分支联系紧密，具有极高研究价值。借助群分次代数如Clifford代数、类八元素代数等特殊群分次扭群代数，研究Hurwitz问题中可容许三元组、Yuzvinsky猜想中可证实位置以及高次型版本Hurwitz问题等。申请人在之前参与的工作可以以扭群代数完整地构造出Hurwitz问题中以其他方法得到的所有已知结果，可以毫无信息损失地表达Hurwitz问题中的所有例子，因此拟采用此方法继续研究更多例子与不同例子之间的关系；在Yuzvnsky猜想中，在引入颜色频率分析方面已取得很大进展，初步证明了形如[9,16k-9,16k-2]位置的猜想正确性，并打算继续以此方法研究；以高次型直和分解中中心幂零部分radical维度为切入点研究Huriwtz问题的高次型版本。

This project intends to research the Hurwitz problem, Yuzvinsky conjecture and other related problems in compositions of quadratic forms, which are closely related to many branches of mathematics and have high research value. With the help of special group graded twisted group algebras such as Clifford algebras and Octonion, the admissible triple in Hurwitz problem, the provable position in Yuzvinsky conjecture and the composition of Hurwitz-like higher form are studied. In the previous work, the applicant can completely construct all the known results obtained by other methods in the Hurwitz problem with the twisted group algebra, and all the examples in the Hurwitz problem can be expressed without information loss, so it is proposed to use this method to continue to study the relationship between more examples and different examples. In the Yuzvnsky conjecture, great progress has been made in introducing color frequency analysis, which preliminarily proves the correctness of the conjecture in the position of [9,16k-9,16k-2], and intends to continue to study it with this method; take the radical dimension of the central nilpotent part of the higher order direct sum decomposition as a starting point to study the Huriwtz problem of the higher forms version.