Lagrange's theorem and integrality for finite commutative hypergroups with applications to strongly regular graphs
The purpose of this paper is to show that algebraic concepts like integrality of coefficients may lead to new insights into the structure of finite hypergroups, and which consequences such results may have to related, more classical algebraic or combinatorial structures. The author defines a finite signed hypergroup \(K\) as an associative \(*\)-algebra over \(\mathbb{C}\) with basis \(K= \{c_0, c_1, \dots, c_n\}\) and properties similar to those of K. Ross or M. Rösler for ``topological'' hypergroups. Moreover, following R. I. Jewett, he requires for a finite hypergroup in addition that all structure constants are positive. It is now standard to define the Haar weights \(\omega (c_i)\) normalized by \(\omega (c_0) =1\) and to develop basic concepts of harmonic analysis for finite commutative signed hypergroups \(K\). This culminates in the fact that the dual \(\widehat {K}\) is again a finite commutative signed hypergroup with \(K^{\wedge \wedge} \simeq K\). A further section is devoted to subhypergroups \(D\) and quotient hypergroups \(K/D\) which can be formed in the finite case always in the natural way. These concepts lead to the following Lagrange-type result: If \(D\) is a subhypergroup of a finite commutative hypergroup such that all weights \(\omega (c_i)\) are integral for both \(K\) and \(\widehat {K}\), then one has \(\omega (D), \omega (K)\in \mathbb{N}\) with \(\omega (D) \mid \omega (K)\). Conjugacy class hypergroups \(K(G)\) of finite groups \(G\) motivate to say that a finite signed hypergroup is integral if the modified structure constants satisfy \[ {{\omega (c_i) \omega (c_j)} \over {\omega (c_k)}} n^k_{i,j}\in \mathbb{Z} \] for all \(i\), \(j\), \(k\). Unfortunately, character hypergroups \(K(\widehat {G})\) of finite groups \(G\) can fail to have this nice property. This observation leads the author to a modification of this integrality called resonance. It is shown that \(K(G)\) and \(K(\widehat {G})\) are resonant in this sense for all finite groups \(G\). The remaining sections are devoted to examples. In particular, all resonant commutative hypergroups \(K\) of order 3 with \(\widehat {K}\) having integral Haar weights are determined completely. This is used to see well-known conditions on the parameters of strongly regular graphs under a new light (as these graphs always admit an associated resonant hypergroup of order 3). The paper concludes with some examples coming from Grassmannians over finite fields.
scientific article; zbMATH DE number 917722