Heronian friezes and Pl\"ucker relations

TL;DR

This paper explores the algebraic relations in Heronian friezes using Plücker relations in Grassmannians, revealing new connections and determinant properties of subfriezes.

Contribution

It introduces novel relations among Heronian frieze entries via Plücker relations and links subfriezes to Plücker friezes, showing certain determinants vanish.

Findings

Relations among Heronian frieze entries derived from Plücker relations

Connection established between subfriezes and Plücker friezes

Determinants of matrices from subfriezes are shown to be zero

Abstract

In this article, we use Pl\"ucker relations in the Grassmannian $Gr(3,n)$ to give relations that hold amongst some of the entries of the Heronian frieze of order $n$. Furthermore, we make a connection between certain subfriezes of a Heronian frieze and Pl\"ucker friezes $P(3,n)$, and then show that some determinants of the matrices whose elements lie in those subfriezes are vanishing.