WordPress and Wikipedia

Every so often, I go to my WordPress blog and read a random item, then step forward or back to get a picture of what was going on in my life then.

Some of this is real feelgood stuff, such as the support I had from students when I was evicted from my office in Queen Mary at two weeks’ notice over Christmas while I was still teaching. It was made clear that, even if my bosses didn’t think so well of me, my students did.

I also come across interesting things which I had forgotten, such as the mythical root system E_7.5. I have opened a web page to go with the ADE book, where I intend to put new facets of the ADE story as I come across them; so I took some of my blog post and put it onto the web page.

Sometimes I reach links that take me to interesting places. But sometimes the randomness comes from elsewhere.

For example, I very seldom look at my own Wikipedia page, but a conference at which I am speaking had linked my name to this page, so I thought I should take a look.

I was delighted to find that a link had been added to this page to the theorem I proved with Dima Fon-Der-Flaass about IBIS groups, those permutation groups in which all irredundant bases have the same size. They have two equivalent characterizations: the irredundant bases are preserved by re-ordering, and they form the bases of a matroid.

However, when I followed the link, I had a bit of a shock. I suspect that it had been written by someone who is not a native English speaker. (Looking at the history, the last edit was made by someone who does not exist, and the one before by a bot.) I am very grateful to whoever put up this page, but I worry that it doesn’t do a great job of communicating the result.

I regard editing references to myself on Wikipedia as a big no-no: this is what prominent people who are trying to keep emnbarrassing information about themselves out of the public eye do.) Correcting factual errors is probably just about OK, but if it is simply page whose information is difficult to extract, I don’t know.

What do you think? Does anyone want to fix the page?

Incidentally, one of Dima’s best insights on this is that there is a wide-ranging generalisation. Let (A_i) be a family of subsets indexed by a set I. A base for the family is a sequence (i₁,…,i_r) of indices such that the intersection of the sets indexed by the sequence is the same as the intersection of all the sets, and no set with index in the sequence contains the sets indexed by its predecessors. Now the three conditions for IBIS groups hold in this generality. Moreover, every matroid can be represented by an IBIS family of sets. (This is an exercise, which I think was due to me, but when I came to prove it just now I found it a bit of a struggle.)

Thus an IBIS group is a group with a conjugacy-closed IBIS family of subgroups whose intersection is trivial (take the peermutation action on the cosets of one subgroup from each conjugacy class). Not every matroid is represented by an IBIS group, indeed we don’t yet have a full characterisation of these, as far as I know.

But this suggests an open problem. Take your favourite variety of algebras, and ask which of them have IBIS families of subalgebras. Rings, or semigroups, anyone?