On Symmetries of Finite Geometries

TL;DR

This paper explores the symmetries of finite geometries through the isospectral sets of Dirac matrices, connecting spectral properties with geometric symmetries and integrable flows.

Contribution

It introduces a framework linking Dirac matrix spectra to finite geometry symmetries and discusses Lax deformations in this context.

Findings

Isospectral Dirac matrices define symmetry subsets of orthogonal/unitary groups.

Lax deformations produce commuting flows on symmetry spaces.

Analogies with Toda systems are established.

Abstract

The isospectral set of the Dirac matrix D=d+d* consists of orthogonal Q for which Q* D Q is an equivalent Dirac matrix. It can serve as the symmetry of a finite geometry G. The symmetry is a subset of the orthogonal group or unitary group and isospectral Lax deformations produce commuting flows d/dt D=[B(g(D)),D] on this symmetry space. In this note, we remark that like in the Toda case, D_t=Q_t* D_0 Q_t with exp(-t g(D))=Q_t R_t solves the Lax system.