Page 107 - Textos de Matemática Vol. 46

A NOTE ON INVARIANT FACTORS 97
This was also proved in [10] in a form valid for EDDs.
But what about the huge family of divisibility relations, mentioned in the
previous section, valid for invariant factors of products of matrices over PIDs (and which actually give the complete answer to the product problem in that setting)? Extending those results to EDDs presents an interesting challenge, necessitating a change in the proofs.
And it would bring an added bonus, since, by a localization argument due to Krull [9], any divisibility relation generally valid in EDDs actually generalizes to GCD domains, rings where every finite collection of elements has a gcd inside the ring. This technique was essentially already used by Kaplansky in [5].
References
[1] R. Bhatia, Linear algebra to quantum cohomology: the story of Alfred Horn’s inequali- ties, Amer. Math. Monthly 108 (2001), 289-318.
[2] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bul- letin AMS 37 (2000), 209-249.
[3] A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225-241.
[4] M. Kapovich, B. Leeb, J. Millson, The generalized triangle inequalities in symmetric
spaces and buildings with applications to algebra, Memoirs AMS 192, no. 896 (2008).
[5] I. Kaplansky, Elementary divisors and modules, Transactions AMS 66 (1949), 464-491.
[6] T. Klein, The multiplication of Schur functions and extensions of p-modules, J. London
Math. Soc. 43 (1968), 280-284.
[7] A. Klyachko, Stable bundles, representation theory and hermitian operators, Selecta
Mathematica 4 (1998), 419-445.
[8] A. Knutson and T. Tao, The honeycomb model of GLn(C) tensor products I: proof of
the saturation conjecture, Journal AMS 12 (1999), 1055-1090.
[9] W. Krull, Allgemeine Bewertunstheorie, J. Reine Angew. Math. 167 (1932), 160-196.
[10] J. F. Queir´o, Invariant factors as approximation numbers, Linear Algebra Appl. 49 (1983), 131-136.
[11] A. P. Santana, J. F. Queiro´ and E. Marques de Sa´, Group representations and matrix spectral problems, Linear and Multilinear Algebra 46 (1999), 1-23.
[12] H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Di↵eren- tialgleichungen, Math. Ann. 71 (1912), 441-479.
(C. Caldeira and J. F. Queir´o) CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
E-mail address: caldeira@mat.uc.pt; jfqueiro@mat.uc.pt