On $p$-schemes of order $p^3$

TL;DR

This paper classifies $p$-schemes of order $p^3$ with a specific thin residue, showing conditions for Schurian property, and constructs non-Schurian schemes algebraically isomorphic to Schurian ones.

Contribution

It provides a classification of $p$-schemes of order $p^3$ based on the structure of their thin residue and constructs examples of non-Schurian schemes with specific algebraic properties.

Findings

$(X,S)$ is Schurian if $T ot eq p^2$

Induces a partial linear space when $T eq p^2$ and $T ot ightarrow C_{p^2}$

Constructs non-Schurian schemes algebraically isomorphic to Schurian schemes

Abstract

Let $(X,S)$ be a $p$-scheme of order $p^3$ and $T$ the thin residue of $S$. Now we assume that $T$ has valency $p^2$. It is easy to see that one of the following holds: (i) $|T|=p^2$ and $T\simeq C_{p^2}$; (ii) $|T|=p^2$ and $T\simeq C_p\times C_p$; (iii) $|T|<p^2$. It is known that $(X,S)$ is Schurian if (i) holds. If (ii) holds, we will show that $(X,S)$ induces a partial linear space on $X/T$. Moreover, the character degrees of $(X,S)$ coincide with the sizes of the lines of the partial linear space. Under the assumption (iii) we will show a construction of non-Schurian $p$-schemes which are algebraically isomorphic to a Schurian $p$-scheme of order $p^3$.