This morning, Esther Beneish

arxived the paper The center of the generic algebra of degree p that may contain the most

significant advance in my favourite problem for over 15 years! In it she

claims to prove that the center of the generic division algebra of

degree p is stably rational for all prime values p. Let me begin by

briefly explaining what the problem is all about. Consider one n by n

matrix A which is sufficiently general, then it will have all its

eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely

determined upto conjugation (that is, base change) by its

characteristic polynomial. In

other words, the conjugacy class of a sufficiently general n by n matrix

depends freely on the coefficients of the characteristic polynomial

(which are the n elementary symmetric functions in the eigenvalues of

the matrix). Now what about **couples** of n by n matrices (A,B) under

**simultaneous conjugation** (that is all couples of the form $~(g A

g^{-1}, g B g^{-1}) $ for some invertible n by n matrix g) ??? So,

does there exist a sort of Jordan normal form for couples of n by n

matrices which are sufficiently general? That is, are there a set of

invariants for such couples which determine it is freely upto

simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old

result due to James Sylvester saying

that this is indeed the case and that the set of invariants consists of

the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio

did a lot more in his paper. He showed that if you could prove this for

couples of matrices, you can also do it for triples, quadruples even any

k-tuples of n by n matrices under simultaneous conjugation. He also

related this problem to the center of the generic division algebra of

degree n (which was introduced earlier by Shimshon Amitsur in a rather

cryptic manner and for a while he simply refused to believe Claudio’s

description of this division algebra as the one generated by two

_generic_ n by n matrices, that is matrices filled with independent

variables). Claudio also gave the description of the center of this

algebra as a field of lattice-invariants (over the symmetric group S(n)

) which was crucial in subsequent investigations. If you are interested

in the history of this problem, its connections with Brauer group

problems and invariant theory and a short description of the tricks used

in proving the results I’ll mention below, you might have a look at the

talk Centers of Generic Division Algebras, the rationality problem 1965-1990

I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally

settled in 1979 by Ed Formanek and a

year later he was able to solve also the case of couples of 4 by 4

matrices in a fabulous paper. In it, he used solvability of S(4) in an

essential way thereby hinting at the possibility that the problem might

no longer have an affirmative answer for larger values of n. When I read

his 4×4 paper I believed that someone able to prove such a result must

have an awesome insight in the inner workings of matrices and decided to

dedicate myself to this problem the moment I would get a permanent

job… . But even then it is a reckless thing to do. Spending all of

your time to such a difficult problem can be frustrating as there is no

guarantee you’ll ever write a paper. Sure, you can find translations of

the problem and as all good problems it will have connections with other

subjects such as moduli spaces of vectorbundles and of quiver

representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with

Christine Bessenrodt we were

able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of

7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if

you could do it for couples of k by k and l by l matrices and if k and l

were coprime then you could also do it for couples of kl by kl matrices,

so the n=6 case was already done). Or did we? Well not quite, our

methods only allowed us to prove that the center is **stably rational**

that is, it becomes rational by freely adjoining extra variables. There

are examples known of stably rational fields which are NOT rational, but

I guess most experts believe that in the case of matrix-invariants

stable rationality will imply rationality. After this paper both

Christine and myself decided to do other things as we believed we had

reached the limits of what the lattice-method could do and we thought a

new idea was required to go further. If today’s paper by Esther turns

out to be correct, we were wrong. The next couple of days/weeks I’ll

have a go at her paper but as my lattice-tricks are pretty rusty this

may take longer than expected. Still, I see that in a couple of weeks

there will be a meeting in

Atlanta were Esther

and all experts in the field will be present (among them David Saltman

and Jean-Louis Colliot-Thelene) so we will know one way or the other

pretty soon. I sincerely hope Esther’s proof will stand the test as she

was the only one courageous enough to devote herself entirely to the

problem, regardless of slow progress.

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