Symmetric matrices defined by plane vector sequences
Mikiya Masuda
TL;DR
The paper studies a construction that associates a symmetric matrix $A$ to a plane-vector sequence $v$ and derives a combinatorial formula for its signature: $\mathrm{Sign}(A)=4R(v)-S(v)$, where $R(v)$ is the rotation number and $S(v)=1+\sum_{i=0}^n\mathrm{sgn}(\det(v_i,v_{i+1}))$. It provides an explicit expression for $A^{-1}$ as a tridiagonal matrix with entries determined from $v$, and interprets the inverse in a geometric/topological setting by linking $A^{-1}$ to intersection pairings on an associated omnioriented quasitoric orbifold $X$. The paper further proves a general relation, via Poincaré duality, that in a 4-dimensional PD space the matrix formed by cup products of Kronecker-dual classes and the matrix formed by Poincaré-dual cup products are inverses, and it gives a geometric realization of $A$ and $A^{-1}$ through the topology of $X$, with $A^{-1}$ serving as the intersection matrix of the characteristic suborbifolds. Overall, the work connects a combinatorial matrix signature formula to explicit inverse computation and to a rich geometric/topological interpretation in toric/quasitoric settings.
Abstract
Motivated by a work of Fu-So-Song, we associate a symmetric matrix $A$ to a plane vector sequence $v$ and give a formula to find the signature of $A$ in terms of the sequence $v$. When $A$ is nonsingular, we interpret the relation between $A$ and $A^{-1}$ from a topological viewpoint. Finally, we associate an omnioriented quasitoric orbifold $X$ of real dimension four to the sequence $v$ and show that $A^{-1}$ is the intersection matrix of the characteristic suborbifolds of $X$.