Eigenvalues, invariant factors, highest weights, and Schubert calculus

Bulletin of the American Mathematical Society

Volumn 37, Issue 3, 2000, Pages 209-249

  Fulton, William ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034215846     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0273-0979-00-00865-x     Document Type: Article

References (67)

1. S. Agnihotri and C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, math.AG/9712013, Math. Res. Lett. 5 (1998), 817-836. MR 2000a:14066
2. A. R. Amir-Moéz, Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations, Duke Math. J. 23 (1956), 463-476. MR 18:105j
3. J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, math.CO/9901118, J. Amer. Math. Soc. 12 (1999), 1119-1178. CMP 99:15
4. 1 \ S for S a finite set, Ph.D. thesis, University of Chicago, 1999.
5. A. Berenstein and R. Sjamaar, Projections of coadjoint orbits and the Hilbert-Mumford criterion, to appear in J. Amer. Math. Soc., math.SG/9810125.
6. F. A. Berezin and I. M. Gel'fand, Some remarks on spherical functions on symmetric Riemannian manifolds, Amer. Math. Soc. Transl. 21 (1962), 193-238. MR 27:1910
7. P. Biane, Free probability for probabilists, MSRI preprint 1998-040.
8. A. Buch, The saturation conjecture (after A. Knutson and T. Tao), to appear in l'Enseignement Math., math.CO/9810180.
9. D. Carlson, Inequalities relating the degrees of elementary divisors within a matrix, Simon Stevin 44 (1970), 3-10. MR 43:232
10. J. Day, W. So, and R. C. Thompson, The spectrum of a Hermitian matrix sum, Linear Algebra Appl. 280 (1998), 289-332. MR 99f:15009
11. H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, to appear in J. Amer. Math. Soc.
12. J. Deruyts, Essai d'une théorie générale des formes algébriques, Mém. Soc. Roy. Sci. Liège 17 (1892), 1-156.
13. I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988). MR 90i:14009
14. A. H. Dooley, J. Repka and N. J. Wildberger, Sums of adjoint orbits, Linear and Multilinear Algebra 36 (1993), 79-101. MR 95k:22025
15. Ky Fan, On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. Nat. Acad. Sci. USA 35 (1949), 652-655. MR 11:600e
16. M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), 27-31. MR 44:4021
17. S. Friedland, Extremal eigenvalue problems for convex sets of symmetric matrices and operators, Israel J. Math. 15 (1973), 311-331. MR 49:2796
18. W. Fulton, Intersection Theory, Springer-Verlag, 1984, 1998. MR 85k:14004
19. _, Young Tableaux, Cambridge University Press, 1997. MR 99f:05119
20. _, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki 845, June, 1998, Astérisque 252 (1998), 255-269. CMP 99:13
21. W. Fulton and J. Harris, Representation Theory, Springer-Verlag, 1991. MR 93a:20069
22. I. M. Gel'fand and M. A. Naimark, The relation between the unitary representations of the complex unimodular group and its unitary subgroup, Izvestiya. Akad. Nauk SSSR,. Ser. Mat 14 (1950), 239-260 (Russian). MR 12:9i
23. D. Grayson, Reduction theory using semistability, Comm. Math. Helvetici 59 (1984), 600 634. MR 86h:22018
24. G. J. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math. 67 (1982), 333-356. MR 84d:22019
25. U. Helmke and J. Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207-225. MR 96b:15039
26. j de B, C. R. Acad. Sc. Paris 254 (1962), 1559-1561. MR 24:A2583
27. A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620-630. MR 16:105c
28. _, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225 241. MR 25:3941
29. C. R. Johnson, Precise intervals for specific eigenvalues of a product of a positive definite and a Hermitian matrix, Linear Algebra Appl. 117 (1989), 159-164.
30. C. R. Johnson and E. A. Schreiner, The relationship between AB and BA, Amer. Math. Monthly 103 (1996), 578-582. MR 97e:15007
31. S. Johnson, The Schubert calculus and eigenvalue inequalities for sums of Hermitian matrices, Ph.D. thesis, University of California. Santa Barbara, 1979.
32. T. Klein, The multiplication of Schur-functions and extensions of p-modules, J. London Math. Society 43 (1968), 280-284. MR 37:4061
33. S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287-297. MR 50:13063
34. A. A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. 4 (1998), 419-445. MR 2000b:14054
35. _, Random walks on symmetric spaces and inequalities for matrix spectra, preprint, 1999.
36. A. Knutson, The symplectic and algebraic geometry of Horn's problem, math.LA/9911088.
37. n(ℂ) tensor products I: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055-1090. MR 2000c:20066
38. A. Knutson, T. Tao and C. Woodward, Honeycombs II: facets of the Littlewood-Richardson cone, to appear.
39. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Annals of Math. 74 (1961), 329-387. MR 26:265
40. S. Lang, Algebra, Second Edition, Addison-Wesley, 1984. MR 86j:00003
41. L. Lesieur, Les problèmes d'intersection sur une variété de Grassmann, C. R. Acad. Sci. 225 (1947), 916-917. MR 9:304c
42. B. V. Lidskii, The proper values of the sum and product of symmetric matrices, Dokl. Akad. Nauk SSSR 74 (1950), 769-772 (Russian).
43. B. V. Lidskii, Spectral polyhedron of the sum of two Hermitian matrices, Funct. Anal. Appl. 16 (1982), 139-140. MR 83k:15009
44. D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London A 233 (1934), 99-141.
45. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second edition, Clarendon, Oxford, 1995. MR 96h:05207
46. D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Third Enlarged Edition, Springer-Verlag, 1994. MR 95m:14012
47. A. Okounkov, Random matrices and random permutations, math.CO/9903176.
48. L. O'Shea and R. Sjamaar, Moment maps and Riemannian symmetric pairs, to appear in Math. Ann., math.SG/9902059.
49. P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci: the Q̃-polynomials approach, Compositio Math. 107 (1997), 11-87. MR 98g:14063
50. J. F. Queiró and E. Marques de Sá, Singular values and invariant factors of matrix sums and products, Linear Algebra Appl. 225 (1995), 43-56. MR 96e:15012
51. R. C. Riddell, Minimax problems on Grassmann manifolds. Sums of eigenvalues, Adv. in Math. 54 (1984), 107-199. MR 86k:58019
52. A. P. Santana, J. F. Queiró and E. Marques de Sá, Group representations and matrix spectral problems, Linear and Multilinear Algebra 46 (1999), 1-23. CMP 2000:02
53. F. Sottile, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 35-39. MR 2000c:14074
54. T. Y. Tam, A unified extension of two results of Ky Fan on the sum of matrices, Proc. Amer. Math. Soc. 126 (1998), 2607-2614. MR 98m:15032
55. G. P. A. Thijsse, The local invariant factors of a product of holomorphic matrix functions: the order 4 case, Integral Equations Operator Theory 16 (1993), 277-304, 605. MR 94c:47021a; MR 94c:47021b
56. R. C. Thompson, An inequality for invariant factors, Proc. Amer. Math. Soc. 86 (1982), 9-11. MR 83k:15014
57. _, Smith invariants of a product of integral matrices, Contemp. Math. 47 (1985), 401-435. MR 87k:15024
58. _, Invariant factors of algebraic combinations of matrices, Frequency domain and state space methods for linear systems, North Holland, 1986, pp. 73-87. MR 89c:15001
59. R. C. Thompson and L. Freede, On the eigenvalues of a sum of Hermitian matrices, Linear Algebra Appl. 4 (1971), 369-376. MR 44:5330
60. R. C. Thompson and S. Therianos, The eigenvalues and singular values of matrix sums and products. VII, Canad. Math. Bull 16 (1973), 561-569. MR 50:4629
61. _, On a construction of B. P. Zwahlen, Linear and Multilinear Algebra 1 (1973/74), 309-325. MR 49:5044
62. B. Totaro, Tensor products of semistables are semistable, Geometry and Analysis on complex Manifolds, World Sci. Publ., 1994, pp. 242-250. MR 98k:14014
63. H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-479.
64. H. Wielandt, An extremum property of sums of eigenvalues, Proc. Amer. Math. Soc. 6 (1955), 106-110. MR 16:785a
65. B. Y. Xi, A general form of the minimum-maximum theorem for eigenvalues of self-conjugate quaternion matrices, Nei Mongol Daxue Xuebao Ziran Kexue 22 (1991), 455-458 (Chinese). MR 92g:15025
66. A. Zelevinsky, Littlewood-Richardson semigroups, math.CO/9704228, New Perspectives in Algebraic Combinatorics (L. J. Billera, A. Björner, C. Greene, R. E. Simion, R. P. Stanley, eds.), Cambridge University Press (MSRI Publication), 1999, pp. 337-345.
67. B. P. Zwahlen, Über die Eigenwerte der Summe zweier selbstadjungierter Operaten, Comm. Math. Helv. 40 (1965), 81-116. MR 32:7566