Zeta Functions for Spherical Tits Buildings of Finite General Linear Groups

In this paper, we define edge zeta functions for spherical buildings associated with finite general linear groups.

We derive elegant formulas for these zeta functions and reveal patterns of eigenvalues of these buildings, by introducing and applying insightful tools including digraphs X0 and X2, cyclic n-partite graphs, partite-transitive group actions, and Springer's theorem on Hecke algebras.