On infinity type hyperplane arrangements and convex positive bijections

• Published in: Indian journal of pure and applied mathematics Vol. 56; no. 3; pp. 1235 - 1259
• Main Author: Kumar, C. P. Anil
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.09.2025
• Subjects: Hyperplanes; Infinity; Theorems
• ISSN: 0019-5588; 0975-7465
• DOI: 10.1007/s13226-024-00583-7

In this article we prove in main Theorem A that any infinity type real hyperplane arrangement Hnm with the associated normal system N can be represented isomorphically by another infinity type hyperplane arrangement H~nm with a given associated normal system N~ if and only if the normal systems N and N~ are isomorphic, that is, there is a convex positive bijection between a pair of associated sets of normal antipodal pairs of vectors of N and N~. We show in Theorem 7.1 that, if two generic hyperplane arrangements Hnm and H~nm are isomorphic then their associated normal systems N and N~ are isomorphic. The converse need not hold, that is, if we have two generic hyperplane arrangements (Hnm)1, (Hnm)2 in Rm, whose associated normal systems N1 and N2 are isomorphic, then there need not exist translates of each of the hyperplanes in the hyperplane arrangement (Hnm)2, giving rise to a translated generic hyperplane arrangement H~nm, such that, H~nm and (Hnm)1 are isomorphic.