Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Symmetries of Finite Geometries

Oliver Knill

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.

On Symmetries of Finite Geometries

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.
Paper Structure (4 sections, 1 theorem, 4 equations, 2 figures)

This paper contains 4 sections, 1 theorem, 4 equations, 2 figures.

Table of Contents

  1. Finite Geometries
  2. The symmetry space
  3. Discussion
  4. Code

Key Result

Theorem 1

If $\exp(-t g(D_0))=Q_t R_t$, then $D_t=Q^*_t D Q_t$ solves $D' = [B,D]$, with $B=g(D)^+-g(D)^-$.

Figures (2)

  • Figure 1: We see the Dirac matrix $D=d+d^*$ of a randomly chosen simplicial complex $G$ and the Dirac matrix $D_t=d_t+d_t^* = c_t+c_t^* + m_t$ of a deformed complex. Unlike the initial $D=D_0$, which has only off diagonal entries, the deformed matrix $D_t$ has a block diagonal part. If we focus in the new geometry of what we "can see" with exterior derivatives $c_t$ mapping $k$-forms to $(k+1)$-forms, we consider the still equivalent geometry $(G,C_t=c_t+c_t^*,R)$. It now represents an expanded space if we measure with the Connes formula.
  • Figure 2: The rate of change of the norm $||m(t)||$ of the block diagonal matrix $m(t) = \oplus_{k=0}^q m_k(t)$ acting on $l^2(G)=\oplus_{k=0}^q l_2(G_k)$ shows inflation. The graph looks pretty similar for different $G$.

Theorems & Definitions (2)

  • Theorem 1
  • proof