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
- A∗B = B∗A;
- (A∗B)∗(A∗A) = A∗(B∗(A∗A)).
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 A∗B = (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 E_{6}.
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 A∗B = (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 A_{5} acting transitively on 15 points, the stabiliser of a point being the Klein group V_{4}. Because this subgroup is contained with index 3 in A_{4}, 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.
Questions
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.
When I saw Sven Reichard’s nice talk, the presented construction reminded me very much of constructions for new designs or strongly regular graphs from old ones (for a given set of parameters). For designs or strongly regular graphs you do these constructions to show that there are actually more than one or, even better, many examples for a given set of parameters. Is there some way how one can ask the same type of question for Jordan algebras or some related combinatorial object?