Using Gröbner Bases to Investigate Flag Algebras and Association Scheme Fusion

It is instructive to see how the use of Gröbner bases [2] can clarify certain computational and theoretic aspects of various topics. Here the choice of topics come from association schemes and fusion of flag algebras arising from generalized n-gons and 2-(v, k, 1) designs [6]. Three different ways of using concepts related to Gröbner bases are given as a guide for using such concepts in similar types of problems. Often it is the case that combinatorialists have to sort out some form of truth about a given object based on its parameters. This can translate into pages and pages of manipulations of intermediate results involving polynomial equations, with the attendant derivations and justifications. But these problems can usually be easily rephrased in terms of finding common zeros of a system of polynomial equations, and knowing for what values of the parameters this happens. This is exactly what Gröbner bases are meant to accomplish. So Gröbner basis theory can provide direct computational answers without the need for further justification, once problems are phrased in terms of generators (polynomials that should be zero) for an ideal in a multivariate polynomial ring, and the need to know the corresponding varieties (set of common zeros). This should be useful either to those wishing to use Gröbner bases to clarify and/or simplify computations or to those interested in seeing how Gröbner bases can be used in “applied theoretical” settings.