Properties of Commutative Association Schemes Derived by Fglm Techniques

Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can characterize completely the Bose-Mesner algebra in terms of a zero-dimensional ideal. A Gröbner basis of this ideal can be easily derived without the use of Buchberger algorithm in an efficient way. From this statement, some nice relations arise between the treatment of zero-dimensional ideals by reordering techniques (FGLM techniques) and some properties of the schemes such as P-polynomiality, and minimal generators of the algebra. 1 Overview and background. In this paper the author is interested in the development of the connection between the algebraic properties of the Bose-Mesner algebra associated to an association scheme and the combinatorial properties of the last one. This investigation is mainly motivated by the fact that computing the eigenvalues of an association scheme is in general difficult (even numerically). We are interested in applications of the theory of association schemes to the areas of coding theory and design theory [4, 3, 7, 8, 15, 20], therefore, we mainly treat commutative symmetric association schemes. Anyway we do not discard the non commutative case and some directions are showed in the conclusions. Main tools the author proposes to show the above connection is using some reordering techniques from zero-dimensional ideals called FGLM techniques. The main objective is pointing the connection between the computer algebra treatment of zero-dimensional ideals and association schemes. In fact much about metric properties (P-polynomial properties) and Q-properties