This is joint work with Takuya Ikuta. Association Schemes Let me begin with a brief definition of an association scheme. An association scheme of class d consists essentially of pairwise commuting symmetric (0,1)-matrices which sum to the all-ones matrix, satisfying a certain condition. An example is the set of distance matrices of a DRG from the previous talk. In that case, A_i is a polynomial in A_1, so the product is a linear combination of these matrices. That is the linear span of these matrices is closed under multiplication. That is basically the definition of an association scheme. Since these matrices can be simultaneously diagonalized, we may arrange the diagonal entries as column vectors, and form a matrix by removing repeated rows. Then one obtains a square matrix, called the eigenmatrix. But I am mainly concerned in this talk, the d by d submatrix of the eigenmatrix, called the principal part. Clebsch Graph Let me give you a simple example. Consider the finite field with 16 elements, the subgroup of index 3 in its multiplicative group, and the Cayley graph over the additive group. Now it is tempting to make these graphs more homogeneous, by decomposing the second graph into edge subgraphs. Of course that is possible, and one obtains an association scheme of class 3. In this way, all the three graphs are isomorphic, but it is important to consider all of the three strongly regular graphs together, rather than just one of them. You will see the reason shortly. Cyclotomic Schemes Let me introduce a shorthand notation Gamma_S In general, P_0 may have more than two distinct entries, even have irrational entries, so these graphs are not necessarily strongly regular. Uniform Cyclotomy Let us consider a sufficient condition for these graphs to be strongly regular. It is possible to get the Clebsch graph from the 3-class cyclotomic scheme. If you fuse the first n relations, then you obtain a SRG. In fact, you don't have to take the first n relations, but any combination will do. So a large number of SRG can be obtained. Moreover, the converse also holds. Association schemes with that property can be characterized by this property. Among cyclotomic schemes, those which have this property were characterized by B-M-W, and B-W-X. This is different from the original definition of amorphous a.s. but shown by van Dam and Muzychuk that it is equivalent to the original definition. GF(2^12) In a cyclotomic scheme which is amorphous, each relation is SRG. But there are some other way to construct SRG from cyclotomic schemes. Here is an example. If you consider three isomorphic but edge-disjoint SRG, and list their eigenvalues along the common eigenspaces, somewhat mysterious thing happens. Here the eigenvalues are just the sums of corresponding eigenvalues of the three graphs, they are all 17 or -15. If you remember what I did for the Clebsch graph, you might be tempted to split the last relation. That turns out to be possible. This splitting is given by shifting the initial set by multiples of 3. The principal part P_0, however, looks different from the case of Clebsch graph. In the Clebsch case, P_0 has a constant diagonal and constant off-diagonal entries. That is not the case, as we can see from the collapsed matrix. In this case, surprisingly you find an incidence matrix of PG(3,2). Symmetric Designs This is in general true, under some assumptions. Suppose that, in an association scheme, all the nontrivial valencies and all the nontrivial multiplicities are the same. Such an association scheme is called pseudocyclic. Here I don't exclude the trivial 2-design of block size 1, that is its incidence matrix is a linear combination of I and J, we have just seen in the amorphous case. So it's not really a theorem, just a trivial observation. But finding nontrivial examples requires some work. Let me emphasize that none of these fusion is canonical. The resulting scheme is not a cyclotomic scheme. PG(m,q) Projective spaces have special property which other symmetric design may not have, and that property can be used to construct fusion schemes. A projective space has a line, so we may assume that the first q+1 columns represent a line in the projective space. Now fuse all the relations not on that line. Then we have altogether q+2 lines, and in this way, we obtain an association scheme of class q+2. We can apply this construction to the known example, namely the 15-class scheme on 2^12 points. That was on PG(3,2) which has line size 3, so we obtain a 4-class scheme. This is the scheme discovered by van Dam, as the first primitive counterexample to Andrei Ivanov's conjecture. The same construction works, so we obtain another two 4-class schemes. Spreads in PG(3,q) When the associated projective space is of dimension 3, then using a spread in PG(3,q), we can fuse relations differently. If P_0 is an incidence matrix of PG(3,q), then we can fuse the relations according the decomposition given by a spread. There is no reason that the full collineation group induces isomorphism of graphs. In fact it does not. In this way, we have amorphous association scheme having the same parameters as, but not isomorphic to, the cyclotomic scheme. Cyclotomic SRG So far, I considered mysterious fusions of cyclotomic schemes. But I realized that I should consider first the cyclotomic schemes themselves, before trying to find fusions. So let us come back to the cyclotomic schemes, and associated Cayley graphs. Here I consider only cyclotomic graphs in the narrow sense, that the graphs is a Cayley graph with respect to a subgroup, not a union of a cosets. The natural problem is to determine when such a cyclotomic graph is strongly regular. Please correct me if I am wrong, but I think this problem is still open. So I decided to do an exhaustive search up to 10^8. We have of course amorphous case, but that is not all. We have five more exceptional cases. Here the design means the one defined by the principal part of the eigenmatrix. So the number of points of the design is the number of classes of the cyclotomic scheme. Only the smallest one I was able to find in the literature. Berlekamp, van Lint and Seidel constructed this SRG as the coset graph of the ternary Golay code. Delsarte mentioned a class 3 fusion scheme of this cyclotomic scheme, in terms of the ternary Golay code. Van Lint and Schrijver identified this graph coming from the ternary Golay code is actually a cyclotomic SRG. So that's what I found for the first graph. But I don't know if the remaining four were known, neither do I know if there are any other cyclotomic SRG. A comment from Qing Xiang: A more complete list of non-amorphous cyclotomic strongly regular graphs can be found in White and Schmidt, Finite Fields and Their Applications 8 (2002), 1--17.