In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, prove that they have a Smith normal form over $\mathbb Z[q]$. particular, examine the arrangement regular $n$-gon plane dihedral model space Platonic polyhedra. each case, matrix associated has Z[q]$ realize their congruent transformation as well.

We consider the quadratic form of a general high-rank deterministic matrix on eigenvectors an N×N Wigner and prove that it has Gaussian fluctuation for each bulk eigenvector in large N limit. The proof is combination energy method Dyson Brownian motion inspired by Marcinek Yau (2021) our recent multiresolvent local laws (Comm. Math. Phys. 388 1005–1048).