On a huge family of non-schurian Schur rings

TL;DR

This paper generalizes Wielandt's classic non-schurian Schur ring from $H\cong \mathbb{Z}_p^2$ to elementary abelian groups of even rank by constructing Schur rings $\mathcal{S}(\Pi)$ from partitions of the line set $\mathcal{L}$ in $V=\mathbb{F}^2$. It encodes slopes via $\mathcal{M}(\Pi)$ and analyzes schurity through the action of $\mathrm{GL}(2,q)$ on lines, together with a key $\ ext{F}_p$-linear lemma. The main result shows that if $\{\infty,0,1\}\subset \mathcal{M}(\Pi)$ and $\mathcal{M}(\Pi)\setminus\{\infty\}$ is not a subfield of $\mathbb{F}$, then $\mathcal{S}(\Pi)$ is non-schurian, yielding a huge family of non-schurian Schur rings for even rank (excluding small exceptional orders). This extends the landscape of Schur ring non-schurity and connects to cyclotomic/amorphous association schemes arising from line-partitions.

Abstract

In his famous monograph on permutation groups, H.~Wielandt gives an example of a Schur ring over an elementary abelian group of order $p^2$ ($p>3$ is a prime), which is non-schurian, that is, it is the transitivity module of no permutation group. Generalizing this example, we construct a huge family of non-schurian Schur rings over elementary abelian groups of even rank.

Theorems & Definitions (8)

• Theorem 2.1
• proof
• Remark
• Theorem 2.2
• Remark
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:nonschurian']}