Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

Bamberg, John¹ ; Lansdown, Jesse¹

¹ Centre for the Mathematics of Symmetry and Computation Department of Mathematics and Statistics The University of Western Australia, W.A., Australia

Algebraic Combinatorics, Tome 6 (2023) no. 1, pp. 197-212

Résumé

In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega ,ℛ)$, and the Krein parameters $q_{i,j}^h$ vanish for some $h\notin T$ and all $i,j\notin T$ ($i,j,h\ne 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega ,ℛ)$ or it is a $T^{\prime }$-design, where $|T^{\prime }|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s,s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d,q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d\ge 3$; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in ${\mathsf{Q}}^{-}(2n-1,q)$, $n⩾3$.

Détail

Reçu le : 2021-07-16
Révisé le : 2022-04-26
Accepté le : 2022-05-05
Publié le : 2023-02-24

DOI : 10.5802/alco.246

Classification : 05E30, 05B30, 05B25, 51E12, 51E05
Keywords: association schemes, Delsarte designs, Krein parameters, hemisystems, $m$-ovoids, generalised octagons, finite geometry

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits