Page 13 - Textos de Matemática Vol. 39

E. MARQUES DE SA´ AND THE INTERLACING INEQUALITIES 3
where they defined a concept which I later called “CS space”. The definition is as follows. Let V be a nonempty set and n a nonnegative integer. To each inte- ger i (1 ≤ i ≤ n) we associate a family Si of subsets of V , called fundamental subsets of dimension i. We can define dimension for any subset of V by
d(A)= min{i:∃S∈Si,S⊇A}. The following axioms are assumed:
(1) V and ∅ are fundamental subsets, with dimensions n and 0 respectively. (2) No fundamental subset is contained in another fundamental subset of
lower dimension.
(3) If A and B are fundamental subsets then
d(A ∩ B) + d(A ∪ B) ≥ d(A) + d(B).
There are many examples of “CS spaces”, although a systematic study has not yet been carried out. For example, a matroid M can be seen as a CS space. Let r(.) be the rank function of the matroid. If A is a subset of M, we define d(A) by
d(A) =| A | − r(A).
If we take as the family Si the family of the subsets B of M that satisfy d(B) = i, we get a “CS space”.
Carlson and Sa´ next consider certain mappings from a “CS space” to a partially ordered set, and to those mappings they associate certain invariants. Using that, they prove interlacing inequalities which contain as particular cases the known inequalities for matrices. That is, the various interlacing theorems are reduced to just one. Actually, things work reasonably well only for the necessity part. But the interlacing inequalities are also sufficient. This means that, in the case of the Sa´-Thompson inequalities, if A11 is known and the interlacing inequalities hold, we can guarantee the existence of a completion of the matrix such that A has f1,...,fn+p as invariant polynomials. It’s in the sufficiency part that, as far as I can see, Carlson and S´a’s theory is not enough. I leave this challenge here.
The curious reader may inquire why the inequalities carry Thompson’s name as well as S´a’s. The answer is simple. They were discovered independently by both, more or less simultaneously, and with very different proofs, in fact completely different in a crucial point. The two papers were published in 1979 in the same journal, Linear Algebra and its Applications. Sa´’s paper was received at LAA on May 24, 1977, and Thompson’s a couple of months later.
I am familiar with Sa´’s case because I was his PhD supervisor. I think he obtained his result in 1975. I returned to the University of Coimbra at the beginning of the 1975-76 academic year, and I received a letter from Marques de Sa´, who had stayed in Lisbon, still in 1975 or maybe in 1976, in which he told me what he had just discovered. At the time, I was following closely the post-revolution events and I wasn’t much interested in anything else. But I read