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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Position at St Andrews

We have just advertised a lectureship in Pure Mathematics. The advertisement is here.

Come join us!

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Combinatorics at Strathclyde

Two years ago, we enjoyed a successful British Combinatorial Conference at the University of Strathclyde in Glasgow. For me it was memorable for several reasons: the appearance of my book “Notes on Counting”, my fall to the floor during the ceilidh, the booksale, and (more seriously) some fine lectures including Graham Farr’s lecture commemorating Bill Tutte’s centenary.

Now the Combinatorics group at Strathclyde (David Bevan, Sergey Kitaev and Einar Steingrímsson) is under threat.

I have received two documents; one on reshaping the Department of Computer and Information Sciences, signed by the Head of Department; and a response from the three members of the Combinatorics group.

The first document is pretty much what you might expect, with lots of fine words about “emerging vision”, “imperative”, “resources … aligned with opportunities for future growth”. Mathematics finds no place in this emerging vision.
I quote:

Combinatorics is not considered to be of fundamental importance to UG-teaching. More broadly speaking, discrete mathematics is of fundamental importance but this can be covered by many staff (eg MSP, Data Analytics and Cybersecurity staff) in the Department.

And more along the same lines. In particular, the group is castigated for not getting “grants around a million pounds or more”. [How many mathematicians anywhere hold such grants?]

The response is a much better written and argued document. (Mathematicians, after all, have to be clear – it is an important part of the job – so I am not at all surprised by this.)

They point out that the three-member group is one of the very strongest research groups in the department, having produced 35% of the department’s four-star papers in the current REF and earning REF and grant funding of close to a million pounds in the last four years. Moreover, discrete mathematics underpins computer science, and the group (being the departments only experts in the area) have developed courses for this. Members of the group have had important administrative roles in the department, having greatly improved systems for interacting with PhD students (criticised in an earlier report).

Moreover, combinatorics, or discrete mathematics (the terms are closer in meaning than the Head of Department seems to think, and if there is a difference, the group’s expertise is broader than “just combinatorics”) is perhaps the most applicable part of mathematics in the information age.

Last year, the Bond report, titled “The era of mathematics”, highlighted the importance of knowledge exchange in mathematics, argued (with many examples) that all parts of mathematics can have application, and pushed for a big increase in funding for mathematics, especially the training of PhD students and postdocs. The Council for the Mathematical Sciences has set up two committees to push forward with this, one to prioritise the recommendations in the Bond Report, and the other to convince policymakers of their importance. I would have thought that Strathclyde would be well-placed to benefit from this, if it is successful. (But not of course under the current reshaping plan.)

There have, sad to say, been several instances in Britain of universities closing down mathematics or getting rid of mathematicians in other ways. One incident that sticks in my mind, in a case where I was involved, occurred when the head of another department, at the start of an interview with the committee, said “I couldn’t hold up my head to be in a University with no mathematics department”. In another case, mathematics was closed down so that computer science could expand; this computer science department now finds that its main job consists of teaching arts students how to switch on the computer. (I exaggerate, but not too much.)

It seems to me that the Strathclyde proposal is a very short-sighted move, and unlikely to be in the department’s long-term interest. Moreover, there is plenty of evidence that collaboration between mathematicians (either a department of them, or a group in a computer science department) and informaticians can be of enormous benefit to both.

If you feel as I do, you may wish to know that the head of Computer and Information Sciences at Strathclyde is Professor Neil Ghani. I am sure you can find his address; you can probably even guess it. You may also wish to contact higher authorities in the University. These are our friends and colleagues; please help if you can.

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Tuna Altınel

Many of you have probably already heard the news that Tuna Altınel, a mathematician of Turkish nationality working in France, was arrested on a family visit to Turkey. His alleged crimes are signing a petition requesting peace talks, and taking part in a public meeting in Lyon “displeasing to the Turkish government”.

You can find latest news here. This website also explains what you can do to help.

Please do anything you can to call for his immediate release.

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Association schemes for diagonal groups

Sean Eberhard commented on my posts on diagonal groups (see here and here). He is correct; there is an association scheme preserved by the full diagonal group with n factors in the socle; it is non-trivial if n > 2. The details take a few pages to write out but the basic idea is completely fine.

This shows that an AS-free group (one which does not preserve a non-trivial association scheme) must be either 2-homogeneous or almost simple. Clearly every 2-homogeneous group is AS-free; there are almost simple examples, but they are rather strange and no pattern has emerged thus far.

I hope a preprint will go on the arXiv sometime soon. But you read it first here (in Sean’s comments over the last few days).

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London Combinatorics Colloquia 2019

In London last week for the two combinatorics colloquia, at Queen Mary and LSE. The weather was unusually grey and rainy; it seems in retrospect that it is almost always fine and sunny for this event, but I know that this is a trick of memory.

Two days, with six talks on each day; as usual I will only say a bit about my favourites.

The first talk on Wednesday at Queen Mary was by Péter Pál Pach, on Polynomial Schur’s Theorem. The Schur’s theorem here is the one that says, if you colour the natural numbers with finitely many colours, there is necessarily a monochromatic solution of the equation x+y = z. His talk was about the related equation x+y = p(z), where p is a polynomial. He described the case p(z) = z2 in detail; in this case, a pointer to what follows, the statement is true for two colours but false for three or more. In general, there is one case which is easily dispatched, typefied by the polynomial p(z) = 2z2+1; the right-hand side is always odd, so if we colour the even numbers red and the odd numbers blue, there is no monochromatic solution. Excluding this and related things, the same result holds: true for two colours, false for three or more.

Julia Wolf gave a talk which made some interesting connections, about an “additive combinatorics” version of the regularity lemma. The bounds which appear in the regularity lemma are very large (tower functions, as Gowers showed). But there is a graph called the “half graph” such that excluding it makes the proof easy and the bounds polynomial. The graph has vertices xi and yi for 1 ≤ i ≤ k, with xi joined to yj if and only if i ≤ j. This (and its infinite analogue) is a familiar object in parts of combinatorics and model theory; it is the case which must be excluded to get a Ramsey theorem for bipartite graphs, and its exclusion gives what model theorists call k-stability. To get nice results for subsets of finite abelian groups, you have to exclude the half-graph from the “additive Cayley graph” of the group, whose vertices are group elements, joined if their sum lies in the special subset. Julia and her collaborators proved strong results under this hypothesis, at which point the model theorists (including Anand Pillay) got into the act, and there was a race between the two groups.

In the afternoon, a rather low-key but very entertaining talk from Keith Ball. He began by exhibiting the Hadamard matrix of order 4, normalised so that its first row is positive. Any other row has two +s and two −s. The positions of the +s and the −s in the rows give a 1-factorisation of K4. So Keith asked: for which normalised Hadamard matrices of order n (a multiple of 4) is there a 1-factorisation of Kn with a bijection between 1-factors and rows after the first such that the two ends of each edge in the 1-factor have the same entry in the corresponding row? And (different question) for which normalised Hadamard matrices is there a 1-factorisation with a bijection between 1-factors and rows sso that the two ends of each edge in the 1-factor have opposite entries in the corresponding row? He conjectures that it is always so. He showed us the proof of the conjecture for Sylvester matrices (which he called “Walsh matrices”), and certain Paley matrices (the prime p has to have the property that 2 is the square of a primitive root).

His proof for Sylvester was by induction; to get from one to the next you take the Kronecker product with the Hadamard matrix of order 2. I wondered whether the property would be preserved by arbitrary Kronecker products of Hadamard matrices. After the talk, Natalie Behague assured me that it was so, and showed me the simple proof.

The next day, at LSE, I will describe just two of the six talks. The first, by Dhruv Mubayi, was my favourite of the whole two days. He talked about classical Ramsey numbers: colour the k-sets of an N-set red and blue; how large does N have to be to guarantee either a s-set with all subsets red, or a n-set with all subsets blue? He was particularly interested in the “off-diagonal” case where k and s are fixed, and n grows. Typically, upper and lower bounds are known, and they are tower functions of heights k and k−1 respectively. He surveyed the state of the art on this.

But his real interest was a refinement, due to Erdős and Hajnal, which introduces a new parameter t, between 1 and {s choose k}. The question is, how large does N have to be to guarantee either a s-set containing at least t red k-sets, or an n-set all of whose k-subsets are blue? Erdős and Hajnal conjectured that there are “threshold” values for t, at which the value of N jumps from polynomial to exponential, and from exponential to double exponential, and so on for each possible order of the tower. Dhruv and his colleagues have shown the existence of the first threshold, and found its precise value. Dhruv explained very clearly how this is a complicated mixture of randomness and induction, and neither part can be left out.

The other talk that impressed me was by Johannes Carmesin. He started off by telling us, or reminding us, about Kuratowski’s theorem: a graph is planar if and only if it has no K5 or K3,3 minor. It was not until 2006 that Lovász thought to ask about higher-dimensional analogues. Johannes interprets this as asking what are the obstructions to embedding a 2-dimensional simplicial complex into 3-space. He developed a theory of “space minors” for 2-dimensional complexes, rather more complicated than graph minors, and gave an excluded minor characterisation of embeddability in 3-space for a simply connected, locally 3-connected 2-complex. One remarkable feature of the theorem is the proof. You show that if the excluded minors don’t occur, then you can embed the complex in a simply connected 3-dimensional manifold. Now you use Perelman’s solution to the Poincaré conjecture: this manifold must be a 3-sphere, and you are done. And there is much more beside. He also has generalisations of Whitney’s theorems, matroid interpretations of everything, and so on. I don’t usually enjoy topology, but this was a very nice talk!

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