This is a preview of subscription content, log in via an institution to check access.
References
J.L. Alperin and R.B. Bell,Groups and Representations, Springer- Verlag, New York, 1995.
T. Ando, “Totally positive matrices,”Linear Alg. Appl. 90 (1987), 165–219.
A. Berenstein, S. Fomin, and A. Zelevinsky, “Parametrizations of canonical bases and totally positive matrices,”Adv. Math. 122 (1996), 49–149.
A. Berenstein and A. Zelevinsky, “Total positivity in Schubert varieties,”Comment. Math. Helv. 72 (1997), 128–166.
F. Brenti, 1987 “The applications of total positivity to combinatorics, and conversely,” —, 56 pp. 451–473.
F. Brenti, “Combinatorics and total positivity,”J. Combin. Theory, Ser.A 71 (1995), 175–218.
R.A. Brualdi and, “Combinatorial matrix theory,”Encyclopedia of Mathematics and its Applications, Vol. 39, Cambridge University Press, Cambridge, 1991.
A. Cohen, N. Linial, and Yu. Rabinovich, “Totally nonnegative matrices and planar arithmetic circuits,” preprint.
C. Cryer, “The LU-factorization of totally positive matrices,”Linear Alg. Appl. 7 (1973), 83–92.
C. Cryer, “Some properties of totally positive matrices,”Linear Alg. Appl. 15 (1976), 1–25.
E. Curtis, D.V. Ingerman, and J. Morrow, “Circular planar graphs and resistor networks,”Linear Alg. Appl. 283 (1998), 115–150.
C.L. Dodgson,The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces, ed. by F.F. Abeles, Lewis Carroll Society of North America, Silver Spring, MD, 1994.
A. Edrei, “On the generating function of totally positive sequences, II,“J. Anal. Math. 2 (1952), 104–109.
M. Fekete, “Über ein Problem von Laguerre,”Rend.Circ. Mat. Palermo 34 (1912), 89–100, 110–120.
S. Fomin and A. Zelevinsky, “Double Bruhat cells and total positivity,”J. Am. Math. Soc. 12 (1999), 335–380.
S. Fomin and A. Zelevinsky, “Recognizing Schubert cells,” preprint math. CO/9807079, July 1998.
S. Fomin and A. Zelevinsky, “Totally nonnegative and oscillatory elements in semisimple groups,” preprint math. RT/9811100, November 1998.
W. Fulton and J. Harris,Representation Theory, Springer-Verlag, New York, 1991.
D. Gale, “Mathematical entertainments: The strange and surprising saga of the Somos sequences,”Math. Intelligencer 13 (1991), no. 1, 40–42.
F.R. Gantmacher and M.G. Krein, “Sur les matrices oscillatoires,” C. ftAcad. Sci. (Paris) 201 (1935), 577–579.
F.R. Gantmacher and M.G. Krein, “Sur les matrices complè tement non négatives et oscillatoires,”Compos. Math. 4 (1937), 445–476.
F.R. Gantmacherand M.G. Krein,Oszillationsmatrizen, Oszillationskerne und Kleine Schwingungen Mechanischer Systeme, Akademie- Verlag, Berlin, 1960. (Russian edition: Moscow-Leningrad, 1950.)
M. Gasca and J.M. Peña, “Total positivity and Neville elimination,”Linear Alg. Appl. 165 (1992), 25–44.
M. Gasca and J.M. Peña, “Total positivity,QR factorization, and Neville elimination,”SIAM J. Matrix Anal. Appl. 14 (1993), 1132- 1140.
M. Gasca and C.A. Micchelli (eds.),Total Positivity and Its Applications, Mathematics and Its Applications No. 359, Kluwer Academic Publishers, Dordrecht, 1996.
I. Gessel and G.X. Viennot, “Binomial determinants, paths and hooklength formulae,“Adv. Math. 58 (1985), 300–321.
I. Gessel and G.X. Viennot, “Determinants, paths, and plane partitions,” preprint.
S. Karlin,Total Positivity, Stanford University Press, Stanford, CA, 1968.
S. Karlin and G. McGregor, “Coincidence probabilities,”Pacific J. Math. 9 (1959), 1141–1164.
B. Lindström, “On the vector representations of induced matroids,”Bull. London Math. Soc. 5 (1973), 85–90.
P. Littelmann, “Bases canoniques et applications,”Séminaire Bourbaki Vol. 1997/98. Astérisque No. 252 (1998), Exp. No. 847, 5, 287–306.
C. Loewner, “On totally positive matrices,”Math. Z. 63 (1955), 338–340.
G. Lusztig, “Total positivity in reductive groups,”Lie Theory and Geometry: in Honor of Bertram Kostant, Progress in Mathematics 123, Birkhäuser, Boston 1994, pp. 531–568.
G. Lusztig, “Introduction to total positivity,”Positivity in Lie theory: Open Problems, de Gruyter Expositions in Mathematics 26, de Gruyter, Berlin, 1998, pp. 133–145.
J.L. Malouf, “An integer sequence from a rational recursion,”Discrete Math. 110 (1992), 257–261.
E.H. Moore, “Concerning the abstract groups of orderk and1/2 k,”Proc. London Math. Soc. 28 (1897), 357–366.
T. Muir,The Theory of Determinants, 2nd ed., Macmillan, London, 1906, Vol. 1.
A. Pinkus, “Spectral properties of totally positive kernels and matrices,”in [25], pp. 477–511.
G. Ringel, “Teilungen der Ebene durch Geraden oder topologische Geraden,“Math. Z. 64 (1956), 79–102.
I.J. Schoenberg,Selected Papers, Birkhäuser, Boston 1988, Vol. 2.
I.J. Schoenberg, “Über Variationsverminderende lineare Transformationen,“Math. Z. 32 (1930), 321–328.
R.P. Stanley, “Log-concave and unimodal sequences in algebra, combinatorics and geometry,“Ann. NY Acad. Sci. 576 (1989), 500–534.
J.R. Stembridge, “Immanants of totally positive matrices are non- negative,”Bull. London Math. Soc. 23 (1991), 422–428.
E. Thoma, “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe,”Math. Z, 85 (1964), 40–61.
J. Tits, “Le problè me des mots dans les groupes de Coxeter,” 1969Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1., 175- 185, Academic Press, London.
A.M. Whitney, “A reduction theorem for totally positive matrices,”J. Anal. Math. 2 (1952), 88–92.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Fomin, S., Zelevinsky, A. Total Positivity: Tests and Parametrizations. The Mathematical Intelligencer 22, 23–33 (2000). https://doi.org/10.1007/BF03024444
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03024444