A few families of non-Schurian association schemes
Presented by Dr. Štefan GYüRKI
Type: Oral presentation
Association schemes are background combinatorial objects in the area of Algebraic Graph Theory (AGT). The main well-known origin of models for association schemes, the so-called Schurian association schemes, appears from transitive permutation groups by taking their 2-orbits (orbitals). All association schemes of order up to 14 can be obtained in this manner. It turns out that not all association schemes are Schurian. The construction of wide classes of non-Schurian association schemes still creates a challenge for experts in the area of AGT. It is known (from the catalogue of small association schemes) that there are exactly two different non-Schurian association schemes of order 18. First, we will describe them in the terms of biaffine planes, i.e., we are giving a computer-free interpretation of these objects. Second, starting with the automorphism groups of these two schemes and with the corresponding coherent configurations (CC), we arranged a successful computer aided experimentation (for a few initial prime numbers $p$), looking for fusion schemes in the considered CC. Basing on the earned observations we are able to prove the existence of at least four infinite families of non-Schurian association schemes of order $2p^2$, where $p>3$ is a prime. Moreover, with the aid of a computer, especially by the using of the computer packages COCO and COCO II (S.~Reichard), we described (for initial values of $p$) their combinatorial, algebraic and color groups of automorphisms. Also more properties of these schemes were explored in the same computational fashion and later on rigorously proved with the aid of theoretical tools of AGT. We will also show, how these association schemes are related to some well-known objects from the extremal graph theory.
Location: Portorož, Slovenia
Address: University of Primorska, Faculty of Tourism Studies, Obala 11a, SI-6320 Portorož - Portorose, Slovenia