G2D2, 2: first week


The campus of the China Three Gorges Hotel has a small river, full of waterlilies and lotus flowers. Beside it runs a main road, along which big water trucks run, shooting atomised water from cannon-like structures; we are told this is to reduce the air pollution.

After an introductory party on Monday, the conference opened at 9:00 on Tuesday morning. In the first week, much of the activity is minicourses 1 and 2, the first on “Laplacian eigenvalues and optimality” given by Rosemary and me, the second on “Topics in representation theory” by Tullio Ceccherini-Silberstein.

But first, some words on the two logos, which appear at the top of the first post in this sequence.

The meeting is being held in the Three Gorges Mathematics Research Centre, whose logo is on the right. It is part of the China Three Gorges University, near the famous Three Gorges on the Yangtse River. The University’s logo, which I am sure your search engine will quickly find, features three vertical wavy shapes, which presumably are intended to suggest water cascading through the three gorges. The TGMRC logo cleverly replaces these random shapes by integral signs, which they somewhat resemble.

The TGMRC is on a corridor on the top floor of a U-shaped building called L1 or L2, depending on which side you enter. The top of the U is closed by a high bridge. So it somewhat resembles the new location of the ICMS in Edinburgh, except that that building is entirely enclosed, whereas the central courtyard of L1/L2 is open to sun and rain. There is a nice lecture room with comfortable chairs. Although there is no public space except for the corridor (where the welcome party was held), there are a number of rooms which can be used as “breakout rooms” for small discussions.

The conference logo on the left is a Deza graph. There has been a recent upsurge of activity on Deza graphs, which I suspect is driven by Sergey Goryainov.

A Deza graph is a regular connected graph for which the number of common neighbours of two vertices takes just two distinct values. If adjacent vertices have one number of common neighbours and non-adjacent vertices the other, then the graph is strongly regular. So a strictly Deza graph is defined to be a Deza graph of diameter 2 which is not strongly regular. The graph in the G2D2 logo is the smallest strictly Deza graph, as you can easily verify. I do not know whether the colours have any mathematical significance.

On Wednesday, Dmitry Panasenko told us about a clever computer search which found all the strictly Deza graphs on up to 21 vertices. For one parameter set (I think on 20 vertices) there are many non-isomorphic graphs (I think around 20 or 30, though I didn’t copy down the number from a long table). Then Soesoe Zaw told us about some new constructions of strictly Deza graphs from the Berlekamp–van Lint–Seidel graph, using the operation of dual Seidel switching due to Willem Haemers. This pleases me, since I am typing this on a computer called Seidel.

The Berlekamp–van Lint–Seidel graph is a graph on 243 vertices which is dual (in the sense of Delsarte duality for association schemes) to the graph associated with the perfect ternary Golay code C of dimension 5 over the field of 3 elements. Take the dual code of C (which is 6-dimensional of length 11 and contains C as a subcode of codimension 1). The vertices of the graph are the cosets of this dual code; two vertices are joined if their difference contains a word of weight 1. It is a “sporadic” strongly regular graph with 243 vertices, and from it Soesoe was able to construct a strictly Deza graph on 243 vertices and two of them on 486 vertices.

Seidel’s name came up again in Wei-Hsuan Yu’s talk on equiangular lines, in which he began with the results which I saw from close-up in the early 1970s by Lemmens, Seidel, Higman, Taylor, Gerzon, and so on, and then gave some improvements, some of which have been obtained by semi-definite programming; and again in Alexander Gavrilyuk’s talk on digraphs whose Hermitian adjacency matrices have spectral radius at most 2.

One of the invited speakers was Gareth Jones. He was going to tell us about his recent theorem that many triangle groups contain uncountably many maximal nonparabolic subgroups (subgroups maximal with respect to containing no parabolic elements in their action on the hyperbolic plane). But for reasons not revealed, he was unable to come to the meeting. Instead, Alexander Mednykh had offered to give a talk to Gareth’s slides. Very brave, I thought. But Sasha began by explaining to us the basic idea behind the proof, a mix of maps and hypermaps, coverings, Riemann surfaces, etc.

Qing Xiang gave us new bounds for the size of a partial ovoid in either the generalised quadrangle E(5,q) (what I might call Ω(6,q) and in the Ree–Tits generalised octagon; in each case, previous bounds were of order a power of q, but he and his co-authors have managed to lower the exponent of the power. He showed us Chris Godsil’s nice proof of Jef Thas’ q3q2+q bound in the GQ case, based on association scheme techniques (and so potentially useable in much more general situations). The new bound uses modulo p techniques and very specific results about dimensions of modules for algebraic groups, and while much more powerful in this case is likely to be much more specialised.

There have been several mentions of Cayley graphs, and I must point out a notational trap. Given a subset S of a group G, not containing the identity, the Cayley (di)graph Cay(G,S) is the graph with vertex set G and with an arc from x to sx for every s in S. If S is inverse-closed, it is an undirected graph. It admits the action of G on itself by right multiplication (the right regular action) as automorphisms. In fact, any graph which admits G acting regularly as a group of automorphisms is necessarily a Cayley graph for G.

Now there are two, quite different, definitions of a normal Cayley graph:

  • The definition I learned first says that Cay(G,S) is a normal Cayley graph if S is a normal subset of G (closed under conjugation by elements of G); equivalently, the graph admits both the left and the right regular action of G as automorphisms.
  • The definition which seems to be standard among participants at this conference is: Cay(G,S) is a normal Cayley graph if G is a normal subgroup of the full automorphism group of the graph.

The first definition is very restrictive but also very useful. For two examples,

  • A graph invariant under the primitive group of simple diagonal type whose socle is the product of two copies of the simple group T is a normal Cayley graph for T.
  • For n > 3, Henson’s universal homogeneous Kn-free graph is not a normal Cayley graph for any group (though Greg Cherlin showed that it is a Cayley graph).

The second definition, by contrast, says essentially that a normal Cayley graph has no “unexpected” automorphisms: its automorphism group is contained in the holomorph of G. In accordance with the principle that graphs tend to have few automorphisms other than those you must have, we would expect that most Cayley graphs are normal; this has indeed been conjectured, with some evidence.

I wish that we had different words for these two obviously useful concepts. But it is probably too late for that now; one of them will just fade away.

The paragraph two above reminds me of an old theorem of mine, the citation for which was just imported from Scopus into the St Andrews research repository. For every finite group G, there is a constant α(G), between 0 and 1, so that, in the class of n-vertex graphs whose automorphism group contains a subgroup isomorphic to G, the proportion whose automorphism group is exactly G tends to the limit α(G) as n→∞. In accordance with the above principle, you might expect that most groups have α(G) = 1; but in fact this happens if and only if G is a direct product of symmetric groups, and the values of α are dense in the unit interval.

The idea of terminology fading away came up in Misha Muzychuk’s nice survey of coherent configurations for the summer school. As I have probably pointed out here before, at the end of the 1960s, within a couple of years of each other, Donald Higman invented coherent configurations, while Weisfeiler and Leman invented cellular algebras. The latter notation has faded away, partly because the term has been re-used with an entirely different meaning. But long before this, Bose and Nair invented association schemes, which are an important special case of coherent configurations. There are four levels of generality; in increasing generality, a coherent configuration may be symmetric, commutative, homogeneous, or none of the above. Bose and Nair used the term “association scheme” for symmetric c.c.; Delsarte used it for commutative c.c.; Bannai and Ito used it for homogeneous c.c., and this seems to have become standard. I prefer to use the term “coherent configuration” with appropriate qualification.

Unfortunately Misha did not have time to describe his recent construction, with Klin and Reichard, of proper Jordan schemes; but Mike Kagan will talk later in the meeting, and I think he might say something about this.

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G2D2, 1: setting the scene


I’m in Yichang, at the China Three Gorges University, for the conference and summer school G2D2 (“Groups and graphs, designs and dynamics”).

This is the sixth meeting in the series: the previous ones were

  1. G2C2: Groups and graphs, cycles and coverings, Novosibirsk, Russia
  2. G2A2: Groups and graphs, algorithms and automata, Ekaterinburg, Russia
  3. G2S2: Groups and graphs, spectra and symmetries, Novosibirsk, Russia
  4. G2M2: Groups and graphs, metrics and manifolds, Ekaterinburg, Russia
  5. G2R2: Groups and graphs, representations and relations, Novosibirsk, Russia

The event following this one will be G2G2: Groups and graphs, geometries and GAP, and will be in Rogla and Portorož, Slovenia, as a satellite meeting of the European Congress of Mathematics.

This sequence inevitably invites one to play alphabet games. I would like to say “Groups and graphs, symmetry and structure”, but we have already had G2S2 which included symmetries. So perhaps I will say “G2I2: Groups and graphs, incidence and independence”. Perhaps “G2F2: Groups and Graphs, fields and functions” (or F2 could be the field with two elements).

The meetings are instructional workshops as well as conferences, and Rosemary and I are doing our double act on “Laplacian eigenvalues and optimality”, indeed we start off the meeting at 9am tomorrow morning.

Last week we were in the beautiful town of Hangzhou on West Lake, with its lotus blossoms at the peak of perfection, willow trees overhanging the water on which small boats plied, and pagodas standing above the trees.

West Lake

I gave a short course on “Four precious jewels”: you can find the slides on my St Andrews web page. (I have talked about this before but the short course gave the opportunity for a slightly longer presentation.) Yesterday we took the train from Hangzhou to Yichang, a seven-hour journey.

Things were in some doubt, as Hangzhou was battered by Typhoon Lekina, which brought heavy rain and strong winds over the previous two days; but the morning dawned fine and clear with hardly a breeze, and everything worked fine.

I hope to report on the meeting later, firewalls and slowness of response permitting.

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Some delay expected

In a few hours I will start out on a month-long trip to China.

I am not sure whether I will have the opportunity to post anything new or deal with incoming comments during that time. The Chinese disable several of my crucial work tools.

So please, if you don’t hear anything for a while, just be patient.

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BCC at Birmingham, days 4 and 5

Thursday saw my favourite of the plenary talks. Iain Moffatt told us about ribbon graphs and delta-matroids. Ribbon graphs are essentially the same as graphs drawn on surfaces (we just fatten up each vertex to a disc and each edge to a ribbon), while delta-matroids stand in the same relation to them as matroids do to ordinary graphs. But there was a lot more to the story. You can form the dual of a planar graph, by putting a vertex in each face and an edge crossing each edge of the original graph. Iain invited us to consider dualising with respect to just some of the edges. In the ribbon graph world this is possible, and the name given to the operation (a twist) gives away the idea: ribbons can be given a twist, as in the construction of a Möbius band. There was much more, for example a connection with linear algebra: delta-matroids are represented by symmetric or skew-symmetric matrices, where we look at all the principal submatrices and ask whether they are non-singular. Everything fits together nicely: the operations of twist on ribbon graphs, symmetric exchange (the relevant version of the exchange axiom) for delta-matroids, and pivot for matrices all correspond exactly.

We had the last two talks in the Designs and Latin Squares mini-symposium, with Sergey Goryainov talking about constructions of Neumaier graphs, and Michael Kinyon talking about some latin-square-like objects related to set-theoretical Yang–Baxter such as racks, quandles, and the newly-named rumples.

On Thursday night we had the conference dinner. Our mini-symposium participants and like-minded people had been put together on a table, and we had a very congenial meal. Indeed, participants remarked on the generally high standard of the food at the conference; I found that usually we were so well fed at breakfast, coffee break, lunch and tea break that no dinner was necessary.

The publishers who had had book tables at the meeting (Cambridge University Press and Springer) had donated some of their unsold books to the British Combinatorial Committee (a gesture for which we are very grateful!). On Thursday we sold them off to participants at reduced prices. On Friday in the coffee break, Andrew Treglown and I did a double act as auctioneers and got rid of the remaining unsold books at even further reduced prices. Some people walked away with great bargains.

The last talk was by Gabor Tardos, on ordered versions of Turán theory: our graphs have total orders on the vertex set, and we ask about the maximum number of edges of a vertex-ordered graph G on n vertices which does not contain a given vertex-ordered graph H as a subgraph. As in classical Turán theory, the answer is asymptotically a fraction (1−1/m) of the number of edges in the complete graph, for some parameter m. Classically, m is one less than the chromatic number of H; in the vertex-ordered case, it is one less than the minimum number of parts in a partition of the vertex set of H into edge-free intervals, so a sort of ordered version of chromatic number. Although many puzzles remain, at least this type of chromatic number is easier in some ways than the usual; for example, it is linear-time computable.

My take on this, for what it is worth, is that we are looking at structures with two relations, a graph and a total order; as a graph we take subgraphs (we are allowed to throw away edges), but as an order we must take the induced substructure. This could be generalised to structures with two sets of relations, where substructures may throw away instances of relations in the first set but must be induced substructures of the second set. This includes, for example, permutation patterns, where there are two total orders and we take induced substructures of both.

In the edge-ordered case, again m+1 is a sort of “chromatic number” but there is an element of Ramsey theory in this case; it may be larger than the number of vertices of H, and may even be infinite (this happens if arbitrarily large complete graphs can be ordered so as to contain no copy of H, so that the maximum number of edges is the number in the complete graph on n vertices.

In summary, a great conference, very well organised; I look forward to seeing many of the participants at the next BCC, in Durham on 5–9 July 2021.

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BCC at Birmingham, days 1-3

This week I am in Birmingham for the British Combinatorial Conference.

The organisation of the conference is outstanding. For one small example, yesterday, fifteen minutes before the Business Meeting was due to start, the Chairman noticed that we didn’t have the minutes of the previous Business Meeting to approve. The Secretary had the file on a laptop, and before the meeting started we had fifty printed copies to distribute.

After the excitement about ADE last week, these diagrams reappeared twice in the first couple of days, Hendrik Van Maldeghem (who talked about geometrical and combinatorial constructions of buildings) showed us all the crystallographic Coxeter–Dynkin diagrams. In a completely different context, Alexander Gavrilyuk mentioned the fact that connected simple graphs with spectral radius at most 2 are the ADE diagrams and the extended ADE diagrams. He attributed this to Smith (1969) and Lemmens and Seidel (1973). I think it would be not unjust to say that this result was part of the classification of the complex simple Lie algebras by Cartan and Killing in the last decade of the nineteenth century. That aside, Alexander was extending this to directed graphs, using a Hermitian adjacency matrix with entries 1 if there are arcs both ways between two vertices, while single arcs from v to w have i (the complex fourth root of 1) in position (v,w) and −i in position (w,v). This had been done by Guo and Mohar, but some small corrections were necessary; he used results of Greaves and McKee to achieve these. (As a footnote to this, it seems to be that to use τ, a complex 6th root of unity, in place of i would be more natural, since the sum of τ and its complex conjugate is 1 rather than 0.)

In fact, for the graph case, much more is known: the graphs whose greatest eigenvalue is at most 2 are the ADE diagrams and their extensions, but the graphs whose least eigenvalue is at least −2 can also be described.

The conference featured mini-symposia, and I organised one on “Designs and Finite Geometries”, which in my opinion has had some beautiful talks so far, from Ian Wanless on plexes in Latin squares, Rosemary Bailey on designs related to the Sylvester graphs (and the wrong turnings on the way to finding them), Peter Keevash on his and others’ results on existence of designs (including the fact that estimates for the number of Steiner systems, asymptotic in the logarithm, are now available, and hinting that he had constructions of large sets of Steiner systems for large admissible orders), and Moura Paterson on authentication schemes.

One of the most exciting talks was by Igor Pak. He has formulae, and good asymptotic estimates, for the numbers of standard Young tableaux for various skew Young diagrams. This was a mix of all kinds of things, including counting linear extensions of posets, rhombus tilings, plane partitions, counting disjoint paths, Vershik’s limiting tableau shapes, and a remarkable formula of Coxeter, which (if I copied it correctly) says

Σ(φn/n2) cos(2πn/5)  =  π2/100

(the sum over all positive integers n.)

Coxeter’s discovery of this formula was based on the existence of the 600-cell (a regular polytope in 4 dimensions) and some spherical geometry. As far as I can tell, the formula was not actually used in the talk, but the philosophy of it led to some of the things that came later.

Two things about the talk were a pity. First, there was no paper in the Proceedings. (In the history of the BCC, it has happened a few times that a speaker provided no talk; indeed I was the editor of the first “published-in-advance” volume, at Royal Holloway in 1975, where I failed to get papers from either Conway or Kasteleyn.) So I am unable to check these details. Second, Igor started in a bit of a rush, and some things were not clearly explained. For example, I think some nodding acquaintance with Plancherel measure is needed to make sense of the Vershik asympotic shape of a random Young diagram, and I didn’t find that in the talk. But it was so full of amazing stuff that it is perhaps churlish to complain.

Apart from these I will be very selective in my reporting. One contributed talk I really enjoyed was by Natasha Dobrinen, on the Ramsey theory of Henson’s homogeneous Kn-free graphs, which included a description of them in terms of trees. It went part rather fast (the talks were only 20 minutes), but I wonder whether this leads to a probabilistic approach to Henson’s graph. I have reported before how I laboured over this, and how Anatoly Vershik explained to me his construction with Petrov in a leisurely afternoon in Penderel’s Oak in London – a construction which is clearly related to the topic of graphons, the subject of Dan Král’s talk.

Then there was a sequence of three nice talks on quite different topics, but all related to permutations (in the combinatorial rather than the group-theoretic sense). Simon Blackburn proved a nice asymptotic result about random permutations for the uniform measure. At the end, Robert Johnson asked whether there were similar results for other measures. This was because Robert’s talk, which was next, was able to prove some of the results for wider classes of measures, though not for the Boltzmann measure, which he gave as an open problem. Then David Galvin talked. One of his results was that, far from being monotonic, the sequence of coefficients (excluding the constant term) in the independent set polynomial of a graph with independence number m can be any permutation of {1,…m}. This suggested to me another interesting measure on permutations. Choose n much larger than m, and choose a random graph on n vertices with independence number m; this induces a probability measure on the permutations. Does this measure tend to a limit as n→∞? If so, this could claim to be a “natural” measure on permutations. Fred thought this was an interesting question.

Any ideas?

We had a reception in the remarkable Barber Institute of Fine Arts. Guided tours of the gallery were offered. We went upstairs, and the first picture we saw was René Magritte’s famous picture “The flavour of tears”. Tuesday was the concert, and apart from having to move to a different room because the piano hadn’t been unlocked, we had a remarkable evening’s entertainment; there are several outstanding pianists at the conference. Today is the excursion, to the Museum of Black Country Living; but I have work to do …

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Aliens Do Exist

The people from the planet Ade have intercepted radio transmissions from Earth, and have discovered that we know about the Petersen graph

Petersen graph

and the root system E6.

One day, a flying saucer from Ade arrives on Earth and delivers an ultimatum:

Either show me a construction of E6 from the Petersen graph, or I will arrange that Boris Johnson will be the next Prime Minister of the United Kingdom.

Too late, I present the solution the alien is looking for.

In the usual representation of E6, the roots all have length √2, and have inner products in the set {2,1,0,−1,−2}. Clearly inner product 2 means the roots are equal and inner product −2 means that one is the negative of the other. So there are three “non-trivial” values for the inner product. But a graph has only two relations, adjacency and non-adjacency; so I will look for a set of 10 vectors with all inner products 0 and 1, with 1 for adjacency and 0 for non-adjacency in the Petersen graph.

The outer pentagon is easy to deal with. Take five basis vectors e1, …, e5 for R5. Then the vectors e1+e2, e2+e3, …, e5+e1, will do the job: each has inner product 1 with its neighbours in the cycle and 0 with its non-neighbours.

To get the inner pentagram, the first step is to find five vectors, each having inner product 1 with one vector in the outer cycle and 0 with all the others. A bit of experimentation shows that (e1e2+e3e4+e5)/2 has inner product 1 with e5+e1 and 0 with the other four; the rest of the vertices can then be found by cyclic permutation.

But there is a problem: these vectors are too short; their lengths are only √5/2.

The solution is to add a sixth dimension and extend the vectors in this dimension. Take a new basis vector e6, and add ce6 to each of the five. To get length √2, we have to choose c = √3/2. This doesn’t affect inner products with the first five, since e6 is orthogonal to them.

Moreover, by a sort of miracle, we find that the inner product of one of these extended vectors with its neighbours in the cycle is 0, while its inner product with its non-neighbours is 1. This gives us the inner pentagram correctly, so the job is done.

But wait a minute, where is the root system? We have produced vectors of the right lengths, and having the right inner products. But a root system must be closed under reflection in the hyperplane perpendicular to each of its vectors. How to we achieve that?

The answer is simple: we simply close up. Take the reflection of the vectors we have in the hyperplanes perpendicular to each of them; this gives some new vectors. Repeat the procedure with the enlarged set until no new vectors are found. Then we have a root system. A little thought, or a little theory, shows that it is an indecomposable root system in R6 with all roots of the same length, and is not A6 or D6; so it must be E6.

I am writing this at the LMS-sponsored Undergraduate Summer School at the University of Leeds. This consists of several mathematicians giving short courses (three lectures and two tutorial sessions) on different branches of mathematics, and a few other people giving one-off talks (or in my case, two one-off talks). The students are very keen and engaged, and tackle the problems (including challenge questions) with great enthusiasm.

One of this week’s lecturers is Sira Gratz from the University of Glasgow, talking about what readers of this will recognise as one of my favourite topics: what I call the ADE affair. Sira entitled her series “Attractive Diagrams Everywhere”, which she has certainly demonstrated; she did admit in her first lecture that someone interpreted the acronym as “Aliens Do Exist”.

Her selection of topics was (to my great pleasure) almost disjoint from what I would have chosen myself: the three lectures described, in detail, root systems, quivers of finite representation type (Gabriel’s theorem shows that these must be orientations of ADE diagrams), and cluster algebras (Fomin and Zelevinsky showed that the finite-dimensional ones are also described by orientations of ADE diagrams). She explained in each case the wider significance of the ideas.

Anyway, tomorrow I am talking about the (countable) random graph. I had planned to start off with the Petersen graph, to point the contrast between finite and infinite. The Petersen graph is highly symmetric, but very atypical among finite graphs (random finite graphs have no non-trivial symmetry with high probability), whereas the countable random graph (there is only one) is highly symmetric.

But I thought it would be nice to make some connection with Sira’s lectures. Thinking about the Petersen graph and root systems led me to the construction I have just described.

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Proper Jordan schemes exist!

A new field of research has just been created by Misha Klin, Misha Muzychuk and Sven Reichard: proper Jordan schemes.

They answered a question which I posed some time ago (I don’t remember when), about whether such objects exist. I would not be interested in “empty set theory”, but now we know that they do exist, so we can go ahead and study them.

A little background. You can find more here or here (the second of these references gives some further reading).

Jordan algebras

A Jordan algebra is a vector space (here over the real numbers) with a multiplication ∗ satisfying

  • AB = BA;
  • (AB)∗(AA) = A∗(B∗(AA)).

These algebras were introduced by Pascual Jordan as a mathematical foundation for quantum mechanics. While they have not caught on for this purpose, they are used in some parts of statistics, especially for estimation of variance components.

Any associative algebra gives rise to a Jordan algebra, on setting AB = (AB+BA)/2. Certain subsets of matrix algebras are also closed under this product and define Jordan algebrasi. Most significantly, the symmetric matrices form a Jordan algebra. An important theorem, the Jordan–von Neumann–Wigner theorem, asserts that apart from some infinite families arising in this way, the only simpleJordan algebra is an exceptional 27-dimensional algebra related to the octonions and the Lie group E6.

Coherent configurations and Jordan schemes

A coherent configuration is a set C of zero-one matrices satisfying

  • the sum of the matrices in C is the all-one matrix J;
  • the identity is the sum of some of the matrices in C;
  • the set C is closed under transposition;
  • the linear span of C is an algebra (closed under matrix multiplication).

The axioms, of course, have a combinatorial interpretation, which you can work out with sufficient diligence.

The configuration is homogeneous if the second condition is replaced by the stronger version asserting that the identity is one of the matrices in C. If in addition all the matrices in C are symmetric, we speak of an association scheme. (I am aware that different terminology is used by different authors here; I have discussed this elsewhere, but let me use my own preferred conventions here.)

For much more on this, see several talks at last year’s Pilsen conference, which can be found here.

Now the definition of a (homogeneous) Jordan scheme is obtained from that of an association scheme by replacing matrix multiplication by the Jordan product AB = (AB+BA)/2.

It is easy to see that, if we take a homogeneous association scheme and “symmetrise” it (by replacing a non-symmetric matrix in C and its transpose by their sum) is a homogeneous Jordan scheme.

My question, which Klin, Muzychuk and Reichard have answered negatively, was:

Is every homogeneous Jordan scheme the symmetrisation of a homogeneous coherent configuration?

So now it makes sense to say that a homogeneous Jordan scheme is proper if it is not the symmetrisation of a homogeneous coherent configuration, and to develop the theory of such objects, as Klin, Muzychuk and Reichard have begun to do.

The example

Actually they have many examples, but I will briefly describe the first one, based on a presentation by Sven Reichard at the Slovenian graph theory conference today.

Start with the alternating group A5 acting transitively on 15 points, the stabiliser of a point being the Klein group V4. Because this subgroup is contained with index 3 in A4, the group is imprimitive, with five blocks of size 3. So apart from equality, there are two invariant relations forming five ordered triangles and the reverse, and three further symmetric relations.

One of the triangles is the island, and the other four make up the continent. The edges joining the island to the continent are called bridges, and are of three colours (corresponding to the three further symmetric relations). Now swap two of the colours on the bridges, leaving the remaining edges alone. Symmetrising the resulting structure gives a homogeneous Jordan scheme, which is proper.


Here are a few questions which could be looked at.

  • Does the Jordan–von Neumann–Wigner theorem have any relevance to proper Jordan schemes? In particular, is there one whose Jordan algebra involves the exceptional simple Jordan algebra?
  • Given a connected simple graph, think of it as an electric circuit, where each edge is a one-ohm resistor. The effective resistance between pairs of terminals defines a metric, called resistance distance, on the vertex set. This is a refinement of the graph structure, similar to that produced by the symmetric version of Weisfeiler–Leman stabilisation. What is the precise relation between these concepts? (Misha Klin, to whom I owe this post, points out that Misha Kagan and Doug Klein have papers relevant to this question.)

Pascual Jordan

Curiously enough, the name of Pascual Jordan (who introduced Jordan algebras) came up in a completely different context yesterday; I would like to say a bit about him.

Jordan is described as one of the unsung heroes of quantum mechanics. It was in a joint paper by Born, Heisenberg and Jordan that the matrix mechanics approach to quantum mechanics was first published (as opposed to Schrödinger’s wave function approach). It is said that the mathematics of matrices in the paper is Jordan’s work. The Nobel Prize was awarded to Heisenberg, Schrödinger and Dirac. Jordan also invented Fermi–Dirac statistics, but because of an unfortunate publication delay he was beaten into print.

According to the MacTutor biography, the reason for the neglect may have been in part his membership of the Nazi party. He wrote in support of the party, but strongly opposed the more extreme views of Ludwig Bieberbach, who believed that there was a real difference between say “French mathematics” and “German mathematics”, and that teaching “German mathematics” to children would increase their “Germanness” (to put it rather crudely).

Anyway, the context in which Jordan’s name came up was a very entertaining lecture given by one of this year’s St Andrews honorary graduands, Jim Al-Khalili, on the new subject of quantum biology. He pointed out that the first paper ever written on quantum biology was by Pascual Jordan, although nobody took it very seriously at the time. It was generally thought that quantum systems would decohere so rapidly in the messy, hot surroundings of a living cell that no effects would be observed. Jim put the opposite “spin” on it: rather than the environment interfering with they system, we can regard the system as exporting information to the environment. It is possible that European robins detect the Earth’s magnetic field (for navigational purposes) by a quantum effect in the bird’s eye, where the collapse of entanglement gives a signal which can be transmitted to the brain by the optic nerve.

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