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

Bamberg, John; Lansdown, Jesse

doi:10.5802/alco.246

Bamberg, John ¹; Lansdown, Jesse ¹

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

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

Abstract

In this paper we show that if $θ$ is a $T$ -design of an association scheme $(Ω, ℛ)$ , and the Krein parameters $q_{i, j}^{h}$ vanish for some $h \notin T$ and all $i, j \notin T$ ( $i, j, h \neq 0$ ), then $θ$ consists of precisely half of the vertices of $(Ω, ℛ)$ or it is a $T^{'}$ -design, where $| T^{'} | > | T |$ . 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 $DQ (2 d, q)$ , $DW (2 d - 1, q)$ and $DH (2 d - 1, q^{2})$ , for $d \geq 3$ ; (iv) the Penttila–Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila–Williford scheme in $Q^{-} (2 n - 1, q)$ , $n ⩾ 3$ .

Received: 2021-07-16
Revised: 2022-04-26
Accepted: 2022-05-05
Published online: 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

Author's affiliations:

Bamberg, John ¹; Lansdown, Jesse ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Implications of vanishing {Krein} parameters on {Delsarte} designs, with applications in finite geometry},
     journal = {Algebraic Combinatorics},
     pages = {197--212},
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Bamberg, John; Lansdown, Jesse. Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 197-212. doi : 10.5802/alco.246. https://alco.centre-mersenne.org/articles/10.5802/alco.246/

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