Modular Construction of Free Hyperplane Arrangements

Symmetry Integrability and Geometry-Methods and Applications

article

Shuhei Tsujie¹ 

[1] Department of Education, Hokkaido University of Education

关键词: hyperplane arrangement;    free arrangement;    matroid;    modular join;    chordality 2020 Mathematics Subject Classification 52C35;    05B35;    05C22;    13N15;   

DOI  :  10.3842/SIGMA.2020.080

来源: National Academy of Science of Ukraine

【 摘 要 】

In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.

【 授权许可】

Unknown