Research On 2-adic Orlik-solomon Algebra Of N-rank Wheel Graph And An Algorithm For Its I₂⁵

Theory of hyperplane arrangements is a relatively active young research area in modern mathematics.It isinterdisciplinary of algebra,combinatorics,algebraic combinatorics,topology,analysis and so on,which has been applied to wavefront and hypergeometric functions,braids and phases.As a result,arrangement of hyperplanes has received more and more attention in the past thirty years.An arrangement of hyperplanes is a finite collection of codimension one affine subspaces in a vector space of finite dimension.Orlik and Solomon proved the cohomology algebra of the complement of arrangement is isomorphic to Orlik-Solomon algebra.In the past thirty years,the questions about Orlik-Solomon algebra have been concerned by scholars in related fields.Zaslavsky proved that the sum of the dimensions of all homogeneous elements of Orlik-Solomon algebra is equal to the number of regions in complement space of the arrangement in a real vector space.The third rank of the quotients of the lower central series of the fundamental group of the complement was called Falk invariant of the arrangement.In recent years,the relevant research on Falk invariant has also been uninterrupted.Many results and algorithms about Falk invariant of some series of arrangements were shown by Jiang,Guo,Schenck and so on.They answered a question of what kind of conditions should be met by the p-tuple corresponding to the generator in the k—adic Orlik-Solomon ideal proposed by Falk in 2001 from different angles.At present,the research on the calculation formula of the fourth homogeneous element of the 2-adic Orlik-Solomon ideal of the wheel graph has been completed,but there is no record of the higher-order homogeneous element of the ideal.In order to further study the algebraic properties of the corresponding graphic arrangements associated to a wheel graph,we select the 2-adic Orlik-Solomon ideal of the wheel graph as the start point,and studies the 2-adic Orlik-Solomon algebra of corresponding graphic arrangements.The generators in the fifth homogeneous element of the ideal are classified and counted.An algorithm for the dimension of the homogeneous element is formulized and implemented on computer.The conjecture of the relationship between the homogeneous dimension and the edge number of the wheel graph is given based on the calculation results,which would be a step to solving the problems by Falk.