The original purpose of our trip to China was to speak at an international workshop on Bannai-Ito theory at Zhejiang University in Hangzhou, a popular Chinese tourist destination.
Hangzhou is set on an artificial (but centuries old) lake, with islands, causeways, willows, pagodas, temples, boats, and so on. The picture above shows the Lei Feng pagoda, Xiao Ying Zhou Island, the Ling Yin temple, and a small lake in the Botanic Gardens. The first three were in places where we were taken by the generous and friendly students, Zhiwen He and Weicong Li. Xiao Ying Zhou Island is the viewpoint for the famous “Three Ponds Mirroring the Moon” which appeared on the 1 yuan note (now replaced by a coin).
Rosemary and I went a day early to give us a bit more time for sightseeing. Conference registration was on Thursday afternoon, and the talks ran from Friday morning until Sunday lunch. It was a packed schedule, with 28 talks in the two and a half days. I will try to give a brief summary. I don’t know what Bannai–Ito theory actually is, but we heard a lot about association schemes, orthogonal polynomials, and so on.
The schedule said that the first item was an opening ceremony at 8.30 followed by my talk at 9. But China apparently doesn’t go in for long opening ceremonies. At 8.30, Jack Koolen said a few words saying essentially that he was happy we had all come, and then it was announced that we would have the conference photograph. So we all traipsed outside for that, and then back in again. Then someone said, shall we wait until 9 or shall we just get started right away? The latter view prevailed, so without further ado I started my lecture. It is true that at about 9 a small group of students arrived and seemed a bit disconcerted to find the first speaker in full flow.
We had been unable to use the regular conference room due to another booking, so the first day’s talks were in a room which was too small (if everyone came, some had to stand), and where the screen was too low: from the back of the room it was impossible to see the last few lines. Despite this, my talk went OK, with even a few interruptions, I’m glad to say. I realised I had left something important out of the statement of one theorem, so made a mental note to fix the slides (which involved adding a new frame).
My talk was on synchronization and separation in the Johnson association scheme, which I have discussed here recently, so I will say no more.
The second talk was by Akihiro Munemasa, who had a nice idea for an approach to permutation representations which are not multiplicity-free. He supposes that the multiplicities are at most two, and the action is imprimitive, so that, if a character has multiplicity two, then it occurs precisely once in the action on blocks. This means that in the isotypic component affording twice this character, there is a “canonical” submodule affording the component in the action on blocks, and then its perp forms a “canonical” second submodule. He was able to use this to get formulae for the minimal idempotents, and hence develop a Delsarte-type theory.
Sung-Yell Song gave us a brief account of partial geometric designs (or 1 1/2 designs according to Arnold Neumaier), with some examples which went past rather fast.
Takayuki Okuda proved a theorem which he claims has an application to isometric embeddings of compact Riemannian symmetric spaces. The theore says that if you take the “opposition involution” on a root system and produce a graph on the quotient in the obvious way, its independence number is the dimension of its fixed point space. He gave the values of this parameter for all the indecomposable root systems.
Rosemary talked about products of association schemes, especially generalized wreath products and crested products.
Then we had a lunch break, with a huge amount of food. There would have been time to do something after lunch, but I didn’t take the opportunity.
Jason Williford discussed the representation diagrams of association schemes, a concept arising naturally from the notion of Q-polynomial schemes, and gave some background.
Bill Martin talked about joint work with Brian Kodalen on the connectivity of the graphs in an association scheme. The obvious conjectures are that both vertex and edge connectivity should be equal to valency, and the only disconnecting sets of the appropriate sizes should be those which isolate a vertex. They can prove that the condition that the graph obtained by deleting a closed neighbourhood is connected is equivalent to there being no two vertices with the same neighbourhoods.
Yan Zhu talked about the structure of shells in the Hamming schemes (the sets of points at fixed distance from a given point), and the question of what t-designs in these schemes look like. A shell in the Johnson scheme is just the direct product of smaller Johnson schemes, but shells in the Hamming scheme are more interesting. (In Rosemary’s terms, there is “something” nested in a Johnson scheme, but exactly what it is is not so clear.) This was joint work with Eiichi Bannai.
Da Zhao presented a theorem, also joint with Bannai, classifying the association schemes which have spherical embeddings in 3-dimensional Euclidean space: there are just seven of these, the five platonic solids and the line graphs of the cube and dodecahedron.
The final session was the highlight, with talks by Tatsuro Ito and Eiichi Bannai. Ito spoke about “Bannai’s Jugendtraum“, which (he said) was to give an alternative approach to the classification of finite simple groups by means of association schemes. The conjugacy class scheme of a finite group is a commutative scheme which is primitive if and only if the group is simple. If we could characterise up to parameters the schemes that arise, then we would just have to characterise all the finite simple groups by their character tables. This dream was not realised and probably is unlikely to be. But this led to P- and Q-polynomial association schemes, Leonard’s theorem and the connection with orthogonal polynomials, and Terwilliger’s theory; there a classification is in sight, as Ito and Terwilliger are on the case. (Also, of course, it led to the celebrated book by Bannai and Ito.)
Then Bannai spoke. He described how, as a student of Iwahori, he had become very familiar with the commutative square
He worked on permutation representations of the groups on the left of the diagram. This led him and Ito to the theorem on Moore graphs (simultaneous with Damerell), association schmes, and the book with Ito, which of course stimulated much more work. He concluded with some problems he still dreams of solving: a bound on the degree of transitivity not using CFSG; re-doing CFSG using algebraic combinatorics (he admitted that these two are out of reach); a theory of commutative association schemes following that of compact symmetric spaces; and generalising tight t-designs to other contexts. In particular, he would like multivariate versions of P- and Q-polynomial association schemes and of Askey–Wilson polynomials.
Then it was time for the conference banquet, where we ate and ate, and finally
staggered home to bed.
The next day, we had to be early again, since we needed to follow the crocodile to the new conference room (though I had a good idea where it was, and could have found it myself). This was in the Sir Run-Run Shaw building; the door we entered said it was the Business Administration department, though the door on the ground floor called it the Mathematics Department (in common with the building we were in yesterday).
We were in a much bigger room, but still not ideal; the room was not raked, so it was still a bit difficult to see the bottom of the screen; and there was storage space under the desks, which meant that my legs didn’t go under unless I twisted them into an uncomfortable contortion.
Kaishun Wang kicked off the proceedings with a talk on weakly distance-transitive digraphs. This is where the relation of two points specifies their distance in both directions, and it is required that these relations form an association scheme. He had cclassifications in some cases including valency 2 or 3, thin case (all nontrivial intersection numbers at most 1), and partial results on diameter 2.
Hajime Tanaka came from quantum probability theory. Essentially, he looks at the spectral distribution of various families, suitably normalised: e.g. for Hamming, he gets Poisson if q/n tends to a finite limit, or normal if it tends to zero; the corresponding orthogonal polynomials are Charlier and Hermite polynomials. Similarly, Johnson schemes give geometric and exponential distributions and Meixner and Laguerre polynomials.
He went on to look for multivariate analogues. In particular, the limit of the spectral distribution of symmetric products of strongly regular graphs, suitably normalised, gives either bivariate Gaussian, or priduct of Gaussian and Poisson, or bivariate Poisson, depending on various limits; he can describe the orthogonal polynomials also.
Sho Suda found the invariant factors of a skew-symmetric ±1 matrix of order 4n+2, as given by the Ehlich–Wojtas theory. As a corollary, Kharaghani’s example of a matrix of order 66 meeting the determinant bound cannot be equivalent to a skew-symmetric matrix.
Zhi Qiao gave a characterisation of dual polar spaces of sufficiently large
diameter: he requires large cliques (meeting the Delsarte bound) and local
conditions on the association scheme, not the full strength of this.
Wei-Hsuan Yu has been improving bounds for numbers of equiangular lines in Euclidean space, and has resolved a number of unknowns in Seidel’s table. But not everything is known yet. He showed that the 90 lines in 20 dimensions cannot have another line added; this is of course not equivalent to showing that 90 is the maximum. The week after the workshop, he came to SJTU to give a seminar, and went into a bit more detail about the semidefinite programming used.
Ziqing Xiang talked about explicit constructions of spherical designs. The existing constructions for given strength and dimension depend on continuity arguments and so are not explicit; he has managed, in a remarkable construction, to give explicit rational constructions, though the price he pays is that the designs are rather large!
Then the lunch break. After lunch, taking advice from Yaokun Wu and the students, Rosemary and I went to the botanic gardens. After paying the small entrance fee, we went up a path over a hill and down the other side, to an area with small lakes with beautifully coloured leaves on the trees and picturesque buildings around them, together with the other side, car parks and huge restaurants.
Paul Terwilliger started the afternoon with a talk on Leonard triples (a new development from Leonard pairs; I am not sure of its application to association schemes). He has constructed three matrices which he calls “pseudo-intertwiners” for the triple, and took us through the rather complicated algebra necessary to verify their properties.
Suogang Guo used the Terwilliger algebra for the folded cube (which is in this case just the coherent algebra for the point stabiliser in its automorphism group – I don’t know how general this is!) to get improved bounds for codes in this graph. However, he didn’t actually show us an explicit example where his method beats known bounds.
Hiroshi Nozaki was interested in bounding the number of vertices in a connected bipartite graph with given valency and given second eigenvalue. On a similar theme, Jongyook Park talked in some generality about the second eigenvalue of a distance-transitive graph and what it tells you.
Sergey Goryainov talked about perfect 2-colourings (that is, equitable partitions with two parts) of the bilinear forms graph of dimension 2×d over the field of two elements. For such a partition, the 2×2 collapsed adjacency matrix has two eigenvalues, one of them the valency; in the case where the other is the second eigenvalue, he was able to give a complete classification. More on this later.
The day ended with Alexei Zhedanov talking about orthogonal polynomials on the unit circle and their connection with double affine Hecke algebras, things which had been mentioned in other talks. He says that these things have applications in solving certain multivariate eigenvalue problems, such as for (α_{1}R_{1}+α_{2}R_{2}+βI)f = 0, which physicists and others are interested in. The polynomials are the Szegő orthogonal polynomials.
The next morning we started a bit later, since we knew the way, but found when we arrived right on time that Eric Moorhouse had just started. Fortunately we didn’t miss much. Eric gave a beautiful talk on double covers, his tool of first resort for many mathematical problems. He showed us by way of introduction that the cube and dodecahedron are (antipodal) double covers of K_{4} and the Petersen graph, and that there is a repeated double cover from K_{4,4} to the 4-cube to a 4-regular graph of girth 6 which is most of PG(2,4). A similar double extension leads from K_{6} to the icosahedron to a graph which is most of PG(2,5). Also, the folded 6-cube is a double cover of both the 6-valent strongly regular graphs on 16 points.
There are 193 known projective planes of order 25, falling into 99 dual pairs. Eric analysed these, taking the quotient of a plane by an involution (discarding fixed points) and then a double cover of this. He found several examples, none of them new. He is trying 49, but there are millions of planes there! He also used an idea due to Conway to get an efficient isomorphism invariant for things involving permutations (such as loops, nonincidence graphs of projective planes, etc.)
He has used double covers of complete graphs (that is, two-graphs – he used this term) to get an isomorphism invariant for ovoids, and show the nonexistence of ovoids in large odd dimensions over odd characteristic.
The piece de resistance wsa the set of Lagrangians (maximal totally isotropic subspaces) in a symplectic space in odd characteristic. There is a 2-valued invariant of triples of Lagrangians called the Maslov index, which is a two-graph and so gives a double cover. He explained how this is constructed in the case of pairwise disjoint Lagrangians X,Y,Z. Any vector in Z has the form x+f(x), where f is an isomorphism from X to Y; using the fact that Z is t.i. and the form is alternating, you find that B(x,f(y)) = B(y,f(x)), so you get a symmetric bilinear form on X, and take the type of the corresponding quadratic form in the Witt ring. Interesting stuff!
Edwin van Dam classified graphs with an eigenvalue of multiplicity 3 which are partially metric (that is, they generate an association scheme, and the distance 2 relation is a single relation in the scheme. APart from complete multipartite graphs, there are just nine such, with interesting relationships, e.g. the Foster graph is a double cover of the Nauru graph.
Hirotake Kurihara comes from differential geometry. He showed us how a irreducible Hermitian symmetric space has a “scaffold” which is a distance-transitive graph. Such a graph is built on the fixed point set of the isometry s_{x} fixing the point x in the definition of symmetric space. It is uniquely determined by the space, and conversely. Grassmann spaces give rise to Johnson graphs. The two exceptional examples give the Schläfli and Gosset graphs.
Wei Wang talked about the graph isomorphism problem, specifically, is it true that almost all graphs are determined by their spectrum (as Willem Haemers conjectured)? He has partial results when you give yourself the spectra of the graph and its complement. He mentioned two things I didn’t know: W. X. Du has improved Babai’s quasipolynomial bound for graph isomorphism to an ^{c log n} bound; and Tao and others have shown that almost all graphs have a simple eigenvalue.
Finally Jack Koolen gave a lovely talk about representing graphs with least eigenvalue −3 or greater by integral lattices with minimum norm −3. Following Conway and Sloane, a lattice L is called s-integrable if (√s)L is isomorphic to a sublattice of the standard lattice Z^{n}. Thus E_{6}, E_{7} and E_{8} are 2-integrable but not 1-integrable. His main theorem asserts that there is a number K such that a graph with minimum valency at least K and least eigenvalue at least −3 is 2-integrable. (This extends Hoffman’s theorem, which has 2 and 1 in place of 3 and 2.) He pointed out that, although the upper bound in the theorem is huge and can surely be improved, the true bound cannot be smaller than 163, since the strongly regular graph of valency 162 on 275 vertices is not 2-integrable. He conjectures that the main theorem holds also for signed graphs, and that all such graphs and signed graphs are s-integrable for some bounded s (perhaps the correct bound is 4).
Then it was the end of the conference. The organisers were thanked; in fact Tao Feng and his team were out of the room when the thanks began, but were quickly fetched. Rosemary and I went for a coffee, then went for a walk by the lake, on the Su causeway, and on Gushan island, before heading back to the hotel where taxis would be fetched to take us to the station for the train back to Shanghai.