Eigenvalues of association schemes of quadratic forms
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 primitive and has the same intersection numbers as the scheme \(\text{Alt}(n+1,q)\) (of alternate matrices), so \(\text{Qua}(n,q)^{\leq 2}\) is self-dual. However \(\text{Alt}(n+1,q)\) is not isomorphic to \(\text{Qua}(n,q)^{\leq 2}\), but to \(\text{Sym}(n,q)^{\leq 2}\). The Appendix lists the eigenvalues of \(\text{Qua}(n,q)\), \(\text{Qua}(n,q)^{E}\) and \(\text{Qua}(n,q)^{\leq 2}\) for \(n=2,3,4\).
scientific article; zbMATH DE number 5280738