Petersburg Department of Steklov Institute of Mathematics

PREPRINT 8/2001

Characterization of cyclotomic schemes on a finite field and normal Schur rings over a cyclic group

This preprint was accepted Reptember 13, 2001
Contact: I.Ponomarenko

ABSTRACT:
It is well known that in general a cyclotomic scheme ${\cal C}$ on a finite
field~${\mathbb F}$ cannot be characterized up to isomorphism by its intersection
numbers. We show that the set of intersection numbers of some scheme
$\wh{\cal C}^{(b)}$ on the $b$-fold Cartesian product of~${\mathbb F}$ where~$b$
is the base number of the group~${\mathop{\rm Aut}\nolimits}({\cal C})$ forms a full set of invariants
of~${\cal C}$. A key point here is that the scheme $\wh{\cal C}^{(b)}$ can be defined
for an arbitrary scheme~${\cal C}$ in a purely combinatorial way. The proof
is based on the complete description of normal Cayley and Schur rings
(introduced in this paper) over a finite cyclic group. The developed
technique enables us to show that a Schur ring over a cyclic group that is
different from the group ring has a nontrivial automorphism.