Within the axiomatic approach (which goes under the name Majorana Theory) to the Monster group and its 196,884-dimensional algebra the following type of problems arose. Given a matrix which is an element of an association scheme, it is necessary to decide on the positivity of the eigenvaulues and on the multiplicity of the zero eigenvalue. M. Mainardis, C. Franchi and the speaker had proposed a method to deal with each of the idempotents of the association scheme separately. The most striking application of this technique was the proof that the highest degree of a 2A-generated alternating subgroup in the Monster is 12.