Nonsymmetric primitive translation schemes of prime power order

摘要

It is well-known that translation schemes of prime order are exactly the cyclotomic schemes. In this paper, we show that there do not exist any two-, three-, or four-class nonsymmetric primitive translation schemes of prime square order. On the other hand, we find new nonsymmetric four- and five-class association schemes from cyclotomy as fission schemes of certain symmetric three-class association schemes. Moreover, we provide an affirmative answer to the following question raised by Song (J Algebr Comb 5:47–55, 1996): are there any other two-class primitive association schemes that admit symmetrizable fission schemes besides the cyclotomic scheme of index 2 for (q equiv 5 hbox { (mod 8)})? To be more specific, we show that a certain two-class primitive association scheme in the finite field ({mathbb F}_{37^3}) constructed by Feng and Xiang (J Comb Theory Ser A 119:245–256, 2012) admits a four-class fission scheme. This fission scheme is realized as a fusion scheme of the cyclotomic scheme of index 28.