Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Eigenvalues, invariant factors, highest weights, and Schubert calculus


Author: William Fulton
Journal: Bull. Amer. Math. Soc. 37 (2000), 209-249
MSC (2000): Primary 15A42, 22E46, 14M15; Secondary 05E15, 13F10, 14C17, 15A18, 47B07
DOI: https://doi.org/10.1090/S0273-0979-00-00865-X
Published electronically: April 5, 2000
MathSciNet review: 1754641
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of $GL_{n}(\mathbb{C} )$. We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices.


References [Enhancements On Off] (What's this?)

  • [AW] 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
  • [AM] 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
  • [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. CMP 99:15
  • [Be] P. Belkale, Local systems on $\mathbb{P} ^{1}\smallsetminus S$ for $S$ a finite set, Ph.D. thesis, University of Chicago, 1999.
  • [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.
  • [BG] 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
  • [Bi] P. Biane, Free probability for probabilists, MSRI preprint 1998-040.
  • [Bu] A. Buch, The saturation conjecture (after A. Knutson and T. Tao), to appear in l'Enseignement Math., math.C0/9810180.
  • [C] D. Carlson, Inequalities relating the degrees of elementary divisors within a matrix, Simon Stevin 44 (1970), 3-10. MR 43:232
  • [DST] 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
  • [DW] H. Derksen and J. Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, to appear in J. Amer. Math. Soc.
  • [D] J. Deruyts, Essai d'une théorie générale des formes algébriques, Mém. Soc. Roy. Sci. Liège 17 (1892), 1-156.
  • [DO] I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988). MR 90i:14009
  • [DRW] A. H. Dooley, J. Repka and N. J. Wildberger, Sums of adjoint orbits, Linear and Multilinear Algebra 36 (1993), 79-101. MR 95k:22025
  • [F] Ky Fan, On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. Nat. Acad. Sci. USA 35 (1949), 652-655. MR 11:600e
  • [Fi] M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), 27-31. MR 44:4021
  • [Fr] S. Friedland, Extremal eigenvalue problems for convex sets of symmetric matrices and operators, Israel J. Math. 15 (1973), 311-331. MR 49:2796
  • [Fu1] W. Fulton, Intersection Theory, Springer-Verlag, 1984, 1998. MR 85k:14004
  • [Fu2] -, Young Tableaux, Cambridge University Press, 1997. MR 99f:05119
  • [Fu3] -, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki 845, June, 1998, Astérisque 252 (1998), 255-269. CMP 99:13
  • [FH] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, 1991. MR 93a:20069
  • [GN] 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
  • [Gr] D. Grayson, Reduction theory using semistability, Comm. Math. Helvetici 59 (1984), 600-634. MR 86h:22018
  • [He] 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
  • [HR] U. Helmke and J. Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207- 225. MR 96b:15039
  • [HZ] J. Hersch and B. Zwahlen, Évaluations par défaut pour une summe quelconque de valeurs propres $\gamma _{k}$ d'un opérateur $C = A+B$, à l'aide de valeurs propres $\alpha _{i}$ de $A$ et $\beta _{j}$ de $B$, C. R. Acad. Sc. Paris 254 (1962), 1559-1561. MR 24:A2583
  • [H1] A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620-630. MR 16:105c
  • [H2] -, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225-241. MR 25:3941
  • [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.
  • [JS] C. R. Johnson and E. A. Schreiner, The relationship between $AB$ and $BA$, Amer. Math. Monthly 103 (1996), 578-582. MR 97e:15007
  • [J] S. Johnson, The Schubert calculus and eigenvalue inequalities for sums of Hermitian matrices, Ph.D. thesis, University of California. Santa Barbara, 1979.
  • [K] T. Klein, The multiplication of Schur-functions and extensions of $p$-modules, J. London Math. Society 43 (1968), 280-284. MR 37:4061
  • [Kle] S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287-297. MR 50:13063
  • [Kl1] A. A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. 4 (1998), 419- 445. MR 2000b:14054
  • [Kl2] -, Random walks on symmetric spaces and inequalities for matrix spectra, preprint, 1999.
  • [Kn] A. Knutson, The symplectic and algebraic geometry of Horn's problem, math.LA/9911088.
  • [KT] A. Knutson and T. Tao, The honeycomb model of ${GL}_{n}(\mathbb{C} )$ tensor products I: proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), 1055-1090. MR 2000c:20066
  • [KTW] A. Knutson, T. Tao and C. Woodward, Honeycombs II: facets of the Littlewood-Richardson cone, to appear.
  • [Ko] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Annals of Math. 74 (1961), 329-387. MR 26:265
  • [La] S. Lang, Algebra, Second Edition, Addison-Wesley, 1984. MR 86j:00003
  • [Le] L. Lesieur, Les problèmes d'intersection sur une variété de Grassmann, C. R. Acad. Sci. 225 (1947), 916-917. MR 9:304c
  • [L1] B. V. Lidskii, The proper values of the sum and product of symmetric matrices, Dokl. Akad. Nauk SSSR 74 (1950), 769-772 (Russian).
  • [L2] B. V. Lidskii, Spectral polyhedron of the sum of two Hermitian matrices, Funct. Anal. Appl. 16 (1982), 139-140. MR 83k:15009
  • [LR] D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London A 233 (1934), 99-141.
  • [Mac] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second edition, Clarendon, Oxford, 1995. MR 96h:05207
  • [MFK] D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, Third Enlarged Edition, Springer-Verlag, 1994. MR 95m:14012
  • [O] A. Okounkov, Random matrices and random permutations, math.CO/9903176.
  • [OS] L. O'Shea and R. Sjamaar, Moment maps and Riemannian symmetric pairs, to appear in Math. Ann., math.SG/9902059.
  • [PR] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci: the $\widetilde {Q}$-polynomials approach, Compositio Math. 107 (1997), 11-87. MR 98g:14063
  • [QS] 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
  • [R] R. C. Riddell, Minimax problems on Grassmann manifolds. Sums of eigenvalues, Adv. in Math. 54 (1984), 107-199. MR 86k:58019
  • [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. CMP 2000:02
  • [So] F. Sottile, The special Schubert calculus is real, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 35-39. MR 2000c:14074
  • [Ta] 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
  • [Thi] 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
  • [Th1] R. C. Thompson, An inequality for invariant factors, Proc. Amer. Math. Soc. 86 (1982), 9-11. MR 83k:15014
  • [Th2] -, Smith invariants of a product of integral matrices, Contemp. Math. 47 (1985), 401-435. MR 87k:15024
  • [Th3] -, 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
  • [TF] 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
  • [TT1] 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
  • [TT2] -, On a construction of B. P. Zwahlen, Linear and Multilinear Algebra 1 (1973/74), 309-325. MR 49:5044
  • [T] B. Totaro, Tensor products of semistables are semistable, Geometry and Analysis on complex Manifolds, World Sci. Publ., 1994, pp. 242-250. MR 98k:14014
  • [W] H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-479.
  • [Wi] H. Wielandt, An extremum property of sums of eigenvalues, Proc. Amer. Math. Soc. 6 (1955), 106-110. MR 16:785a
  • [X] 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
  • [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.
  • [Zw] B. P. Zwahlen, Über die Eigenwerte der Summe zweier selbstadjungierter Operaten, Comm. Math. Helv. 40 (1965), 81-116. MR 32:7566

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 15A42, 22E46, 14M15, 05E15, 13F10, 14C17, 15A18, 47B07

Retrieve articles in all journals with MSC (2000): 15A42, 22E46, 14M15, 05E15, 13F10, 14C17, 15A18, 47B07


Additional Information

William Fulton
Affiliation: University of Michigan, Ann Arbor, MI 48109-1109
Email: wfulton@math.lsa.umich.edu

DOI: https://doi.org/10.1090/S0273-0979-00-00865-X
Received by editor(s): July 1, 1999
Received by editor(s) in revised form: January 3, 2000
Published electronically: April 5, 2000
Additional Notes: The author was partly supported by NSF Grant #DMS9970435.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society