A four-class subscheme of the association of scheme coming from the action of \(\text{PGL} (2,4^f)\)
type: Item
instance of: scholarly article
title: A four-class subscheme of the association of scheme coming from the action of \(\text{PGL} (2,4^f)\)
zbMATH Open document ID: 0990.05142
author: Hajime Tanaka
published in: European Journal of Combinatorics
publication date: 2002-08-07T00:00:00Z
review text: Let \(\Omega\) denote the set of two-element subsets of the projective line \(\text{PG}(1,q)\), where \(q=2^e\). Then \(G=\text{PGL}(2,q)\) acts naturally on \(\Omega\times \Omega\) and the orbits of \(G\) acting on \(\Omega\times \Omega\) form the relations of the association scheme \(X(G,\Omega)\). Let \(H\) be the stabilizer of \(\{\langle (1,0)\rangle, \langle (0,1)\rangle\}\) in \(G\). Then \(X(G,\Omega)\) is the same as \(X(G,G/H)\), and this association scheme is symmetric. In this paper the order of the intersection of each conjugacy class of \(G\) and each double coset of \(H\) in \(G\), and the decomposition of the permutation character \(1_H^G\) into irreducible characters are determined. As a corollary, the character table of \(X(G,\Omega)\) is calculated. Further, it is proved that the conjectured subscheme of de Caen and van Dam of \(X(G,\Omega)\), \(q=4^f\) is indeed an association scheme (Theorem 4.1).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1740674
zbMATH Keywords: association schemes
zbMATH Keywords: character table
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: eujc.2001.0511
OpenAlex ID: W1985832303
cites work: Subschemes of some association schemes
cites work: Q3218140
cites work: Fissioned triangular schemes via the cross-ratio
cites work: Q3851217
cites work: Character table of a controlling association scheme defined by the general orthogonal group \(O(3,q)\)
cites work: The Representations of GL(3,q), GL(4,q), PGL(3,q), and PGL(4,q)
DOI: 10.1006/EUJC.2001.0511
Recommended article: A subadjunction formula and Moishezon fourfolds homeomorphic to \(\mathbb{P}^ 4_{\mathbb{C}}\)
Recommended article: Association schemes from the action of \(\text{PGL}(2,q)\) fixing a nonsingular conic in \(\text{PG}(2,q)\)
Recommended article: Q3760663
Recommended article: The Classification of Pin4-Bundles over a 4-Complex
Recommended article: On Lee association schemes over \(\mathbb{Z}_4\) and their Terwilliger algebra
Recommended article: Q5449161
Recommended article: Three-class association schemes on Galois rings in characteristic 4
Recommended article: A four-class association scheme derived from a hyperbolic quadric in PG(3,q)
Recommended article: Classification of stable rank 2 reflexive sheaves onP4withC2⩽ 2
Recommended article: The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4
description: scientific article; zbMATH DE number 1740674