I have always believed in the onion theory of personality expressed so eloquently by Ibsen in *Peer Gynt*. Well, I have just grown another layer.

I have been honoured by being elected to membership of CAUL, the Centre for Algebra at the University of Lisbon, where I have had several memorable visits (some of which I have described here).

Now I have a Web presence there. Please visit: there is not much there now, but it will grow and develop!

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## About Peter Cameron

I count all the things that need to be counted.

Completely off-topic, but I happened to see this post and had been meaning to ask you something over email. I hope it isn’t “spam” in this context.

The complex character table of a finite group can be renormalized to yield a unitary matrix (this is one way to get Schur column orthogonality from row orthogonality) and hence, replacing each entry of the matrix with the square of its modulus, one obtains a bistochastic matrix. Have people looked at which matrices can occur, or whether groups giving rise to the same bistochastic matrix must share some other properties, and so forth?

(The indirect motivation: in some work I’ve been toying with, and miserably failing to blog about, I’m looking for lower bounds on a constant that can be associated to the character table, and in one part of the analysis I pass to a minorizing constant which only depends on the bistochastic matrix defined as above. I’m curious to get a better feel for how much information is lost in “ignoring phase” in the character table.)

I don’t know if anyone looked at this. What does the character table tell you about a group? Quite a lot, though not everything, even if you include some additional information like the power maps. But this modified version? Does anyone know? Obviously if the group is abelian you learn nothing except its order.

Thanks, Peter. We at least know the cardinality of the centre (because we know the conjugacy class sizes) and the cardinality of the abelianization (because we know the character degrees). I recall seeing a comment in Isaacs’s book which said that the full character table, even without the power maps, can detect whether or not the group is nilpotent; however I don’t remember whether the proof has any hope of working for the modified version of the character table.

Congratulations!