Giá trị riêng, hệ số bất biến, trọng số cao nhất và giải tích Schubert

Agnihotri, S., 1998, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett., 5, 817, 10.4310/MRL.1998.v5.n6.a10

Arf, Cahit, 1939, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math., 181, 1, 10.1515/crll.1940.181.1

[BDJ]BDJ 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.

[Be]Be P. Belkale, Local systems on ℙ¹∖𝕊 for 𝕊 a finite set, Ph.D. thesis, University of Chicago, 1999.

[BS]BS A. Berenstein and R. Sjamaar, Projections of coadjoint orbits and the Hilbert-Mumford criterion, to appear in J. Amer. Math. Soc., math.SG/9810125.

Berezin, F. A., 1962, Some remarks on the theory of spherical functions on symmetric Riemannian manifolds, Amer. Math. Soc. Transl. (2), 21, 193, 10.1090/trans2/021/07

[Bi]Bi P. Biane, Free probability for probabilists, MSRI preprint 1998-040.

[Bu]Bu A. Buch, The saturation conjecture (after A. Knutson and T. Tao), to appear in l’Enseignement Math., math.C0/9810180.

Carlson, David, 1970, Inequalities relating the degrees of elementary divors within a matrix, Simon Stevin, 44, 3

Day, Jane, 1998, The spectrum of a Hermitian matrix sum, Linear Algebra Appl., 280, 289, 10.1016/S0024-3795(98)10019-8

[DW]DW H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, to appear in J. Amer. Math. Soc.

[D]D J. Deruyts, Essai d’une théorie générale des formes algébriques, Mém. Soc. Roy. Sci. Liège 17 (1892), 1–156.

Dolgachev, Igor, 1988, Point sets in projective spaces and theta functions, Ast\'{e}risque, 210

Dooley, A. H., 1993, Sums of adjoint orbits, Linear and Multilinear Algebra, 36, 79, 10.1080/03081089308818278

Ward, Morgan, 1939, The lattice theory of ova, Ann. of Math. (2), 40, 600, 10.2307/1968944

Fiedler, Miroslav, 1971, Bounds for the determinant of the sum of hermitian matrices, Proc. Amer. Math. Soc., 30, 27, 10.2307/2038212

Friedland, Shmuel, 1973, Extremal eigenvalue problems for convex sets of symmetric matrices and operators, Israel J. Math., 15, 311, 10.1007/BF02787574

Fulton, William, 1984, Intersection theory, 2, 10.1007/978-3-662-02421-8

Fulton, William, 1997, Young tableaux, 35

[Fu3]Fu3 \bysame, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki 845, June, 1998, Astérisque 252 (1998), 255–269.

Fulton, William, 1991, Representation theory, 129, 10.1007/978-1-4612-0979-9

Hopkins, Charles, 1939, Rings with minimal condition for left ideals, Ann. of Math. (2), 40, 712, 10.2307/1968951

Grayson, Daniel R., 1984, Reduction theory using semistability, Comment. Math. Helv., 59, 600, 10.1007/BF02566369

Heckman, G. J., 1982, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Invent. Math., 67, 333, 10.1007/BF01393821

Helmke, Uwe, 1995, Eigenvalue inequalities and Schubert calculus, Math. Nachr., 171, 207, 10.1002/mana.19951710113

Hersch, Joseph, 1962, Évaluations par défaut pour une somme quelconque de valeurs propres 𝛾_{𝑘} d’un opérateur 𝐶=𝐴+𝐵 à l’aide de valeurs propres 𝛼₁ de 𝐴 et 𝛽ⱼ de 𝐵, C. R. Acad. Sci. Paris, 254, 1559

Nakayama, Tadasi, 1939, On Frobeniusean algebras. I, Ann. of Math. (2), 40, 611, 10.2307/1968946

Horn, Alfred, 1962, Eigenvalues of sums of Hermitian matrices, Pacific J. Math., 12, 225, 10.2140/pjm.1962.12.225

[Joh]Joh 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.

Johnson, Charles R., 1996, The relationship between 𝐴𝐵 and 𝐵𝐴, Amer. Math. Monthly, 103, 578, 10.2307/2974670

[J]J S. Johnson, The Schubert calculus and eigenvalue inequalities for sums of Hermitian matrices, Ph.D. thesis, University of California. Santa Barbara, 1979.

Klein, T., 1968, The multiplication of Schur-functions and extensions of 𝑝-modules, J. London Math. Soc., 43, 280, 10.1112/jlms/s1-43.1.280

Kleiman, Steven L., 1974, The transversality of a general translate, Compositio Math., 28, 287

Klyachko, Alexander A., 1998, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.), 4, 419, 10.1007/s000290050037

[Kl2]Kl2 \bysame, Random walks on symmetric spaces and inequalities for matrix spectra, preprint, 1999.

[Kn]Kn A. Knutson, The symplectic and algebraic geometry of Horn’s problem, math.LA/9911088.

Knutson, Allen, 1999, The honeycomb model of 𝐺𝐿_{𝑛}(𝐶) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., 12, 1055, 10.1090/S0894-0347-99-00299-4

[KTW]KTW A. Knutson, T. Tao and C. Woodward, Honeycombs II: facets of the Littlewood-Richardson cone, to appear.

Kostant, Bertram, 1961, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2), 74, 329, 10.2307/1970237

Lang, Serge, 1984, Algebra, 2

Birkhoff, Garrett, 1939, A characterization of Boolean algebras, Ann. of Math. (2), 40, 609, 10.2307/1968945

[L1]L1 B. V. Lidskii, The proper values of the sum and product of symmetric matrices, Dokl. Akad. Nauk SSSR 74 (1950), 769–772 (Russian).

Lidskiĭ, B. V., 1982, A polyhedron of the spectrum of the sum of two Hermitian matrices, Funktsional. Anal. i Prilozhen., 16, 76

[LR]LR D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London A 233 (1934), 99–141.

Macdonald, I. G., 1995, Symmetric functions and Hall polynomials, 2, 10.1093/oso/9780198534891.001.0001

Mumford, D., 1994, Geometric invariant theory, 34, 3, 10.1007/978-3-642-57916-5

[O]O A. Okounkov, Random matrices and random permutations, math.CO/9903176.

[OS]OS L. O’Shea and R. Sjamaar, Moment maps and Riemannian symmetric pairs, to appear in Math. Ann., math.SG/9902059.

Pragacz, P., 1997, Formulas for Lagrangian and orthogonal degeneracy loci; 𝑄̃-polynomial approach, Compositio Math., 107, 11, 10.1023/A:1000182205320

Queiró, João F., 1995, Singular values and invariant factors of matrix sums and products, Linear Algebra Appl., 225, 43, 10.1016/0024-3795(93)00317-S

Riddell, R. C., 1984, Minimax problems on Grassmann manifolds. Sums of eigenvalues, Adv. in Math., 54, 107, 10.1016/0001-8708(84)90039-2

[SQS]SQS 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.

Sottile, Frank, 1999, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc., 5, 35, 10.1090/S1079-6762-99-00058-X

Tam, Tin-Yau, 1998, A unified extension of two results of Ky Fan on the sum of matrices, Proc. Amer. Math. Soc., 126, 2607, 10.1090/S0002-9939-98-04770-4

Nakano, Hidegorô, 1939, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3), 21, 357

Thompson, Robert C., 1982, An inequality for invariant factors, Proc. Amer. Math. Soc., 86, 9, 10.2307/2044384

Thompson, Robert C., 1985, Smith invariants of a product of integral matrices, 401, 10.1090/conm/047/828316

Thompson, Robert C., 1986, Invariant factors of algebraic combinations of matrices, 73

Thompson, Robert C., 1971, On the eigenvalues of sums of Hermitian matrices, Linear Algebra Appl., 4, 369, 10.1007/bf01817787

Thompson, R. C., 1973, The eigenvalues and singular values of matrix sums and products. VII, Canad. Math. Bull., 16, 561, 10.4153/CMB-1973-092-x

Thompson, R. C., 1973, On a construction of B. P. Zwahlen, Linear and Multilinear Algebra, 1, 309, 10.1080/03081087408817031

Totaro, Burt, 1994, Tensor products of semistables are semistable, 242

[W]W H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen, Math. Ann. 71 (1912), 441–479.

Nakayama, Tadasi, 1939, On Frobeniusean algebras. I, Ann. of Math. (2), 40, 611, 10.2307/1968946

Xi, Bo Yan, 1991, A general form of the minimum-maximum theorem for eigenvalues of self-conjugate quaternion matrices, Nei Monggol Daxue Xuebao Ziran Kexue, 22, 455

[Z]Z 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.

Zwahlen, Bruno Peter, 1966, Über die Eigenwerte der Summe zweier selbstadjungierter Operatoren, Comment. Math. Helv., 40, 81, 10.1007/BF02564365