Eigenvalues of association schemes of quadratic forms
type: Item
instance of: scholarly article
title: Eigenvalues of association schemes of quadratic forms
zbMATH Open document ID: 1144.05073
author: Changli Ma
author: Jian-Min Ma
author: Rongquan Feng
author: Yangxian Wang
published in: Discrete Mathematics
publication date: 2008-05-29T00:00:00Z
review text: Let \(\text{Qua}(n,q)\) (resp. \(\text{Sym}(n,q)\)) denote the association scheme of quadratic forms (resp. symmetric bilinear forms) in \(n\) variables over the finite field \(\mathbb F_{q}\). \textit{Y. Wang}, \textit{Ch. Wang}, \textit{Ch. Ma}, and \textit{J. Ma} [J. Algebr. Comb. 17, 149--161 (2003; Zbl 1017.05108)] have shown these schemes are formally dual and primitive. Let \(X_{n}\) denote the set of quadratic forms in \(n\) variables over \(\mathbb F_{q}\), \(C_{k}^{(n)}\) the set of forms of type \(k\) in \(X_{n}\), \(Y_{n}\) the set of \(n\times n\) symmetric matrices over \(\mathbb F_{q}\), and \(D_{i}^{(n)}\) (resp. \(D_{2i^{\ast}}^{(n)}\)) the cogredience class of non-alternate (resp. alternate) matrices of rank \(i\) (resp. \(2i\)) in \(Y_{n}\). Then the eigenvalues of \(\text{Qua}(n,q)\) are given by \(f_{i}^{(n)}(C_{j}^{(n)})\) and \(f_{2i^{\ast}}^{(n)}(C_{j}^{(n)})\), where \(f_{i}^{(n)}\) and \(f_{2i^{\ast}}^{(n)}\) are additive characters of \(X_{n}\) defined by \(D_{i}^{(n)}\) and \(D_{2i^{\ast}}^{(n)}\), respectively. This paper gives recursive formulae for these eigenvalues when \(q\) is a power of \(2\). This is achieved by defining a \(1\)- and \(2\)-extension of a quadratic form, describing how the types of extensions are distributed, and what conditions are needed for a given extension to have a given type (Tables 1--14). From these tables are deduced recursive formulae for the eigenvalues \(f_{r}^{(n)}(C_{j}^{(n)})\) (Theorems 4.2, 4.3, 4.4) and the eigenvalues \(f_{2k^{\ast}}^{(n)}(C_{j}^{(n)})\) (Theorems 4.6, 4.7, 4.8). The authors then describe the fusion schemes (obtained by merging associate classes) \(\text{Qua}(n,q)^{E}\) and \(\text{Sym}(n,q)^{N}\) and show that these are formally dual (Theorem 5.2). By further merging classes, the fusion schemes \(\text{Qua}(n,q)^{\leq 2}\) and \(\text{Sym}(n,q)^{\leq 2}\) are obtained and shown to be formally dual (Theorem 5.3). They conclude by remarking that \(\text{Qua}(n,q)^{\leq 2}\) is pr…
reviewed by: M. Rafiq Omar
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 15A99
zbMATH DE Number: 5280738
zbMATH Keywords: association scheme
zbMATH Keywords: quadratic form
zbMATH Keywords: symmetric bilinear form
zbMATH Keywords: eigenvalue
zbMATH Keywords: fusion scheme
zbMATH Keywords: formally dual
MaRDI profile type: Publication
full work available at URL: j.disc.2007.08.027
OpenAlex ID: W2025379314
cites work: Q3218140
cites work: Q3992965
cites work: Association schemes of quadratic forms
cites work: Problems in algebraic combinatorics
cites work: The automorphism group and the convex subgraphs of the quadratic forms graph in characteristic 2
cites work: Q4310042
cites work: Association schemes of symmetric matrices over a finite field of characteristic two
cites work: Association schemes of quadratic forms over a finite field of characteristic two
cites work: Association schemes of quadratic forms and symmetric bilinear forms
DOI: 10.1016/J.DISC.2007.08.027
description: scientific article; zbMATH DE number 5280738