Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Rietsch, Konstanze^1, 2

¹Mathematical Institute, University of Oxford, Oxford, United Kingdom
²King’s College, University of London, London, United Kingdom

Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 363-392

Résumé

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in $GL_n$ forms a real semi-algebraic cell of dimension $n-1$. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of $GL_n(\mathbb {C})$ relying in particular on the positivity of the structure constants, which are enumerative Gromov–Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson’s which we explain with proofs in the type $A$ case.

Détail

DOI : 10.1090/S0894-0347-02-00412-5

Bibliographie

[1] Abrams, Lowell The quantum Euler class and the quantum cohomology of the Grassmannians Israel J. Math. 2000 335 352

[2] Astashkevich, Alexander, Sadov, Vladimir Quantum cohomology of partial flag manifolds 𝐹_{𝑛₁⋯𝑛_{𝑘}} Comm. Math. Phys. 1995 503 528

[3] Berenstein, Arkady, Fomin, Sergey, Zelevinsky, Andrei Parametrizations of canonical bases and totally positive matrices Adv. Math. 1996 49 149

[4] Bertram, Aaron Quantum Schubert calculus Adv. Math. 1997 289 305

[5] Hebroni, P. Sur les inverses des éléments dérivables dans un anneau abstrait C. R. Acad. Sci. Paris 1939 285 287

[6] Cattani, E., Dickenstein, A., Sturmfels, B. Computing multidimensional residues 1996 135 164

[7] Ciocan-Fontanine, Ionuţ Quantum cohomology of flag varieties Internat. Math. Res. Notices 1995 263 277

[8] Ciocan-Fontanine, Ionuţ On quantum cohomology rings of partial flag varieties Duke Math. J. 1999 485 524

[9] Cox, David A., Katz, Sheldon Mirror symmetry and algebraic geometry 1999

[10] Hebroni, P. Sur les inverses des éléments dérivables dans un anneau abstrait C. R. Acad. Sci. Paris 1939 285 287

[11] Fomin, Sergey, Gelfand, Sergei, Postnikov, Alexander Quantum Schubert polynomials J. Amer. Math. Soc. 1997 565 596

[12] Fomin, Sergey, Zelevinsky, Andrei Total positivity: tests and parametrizations Math. Intelligencer 2000 23 33

[13] Fulton, W., Pandharipande, R. Notes on stable maps and quantum cohomology 1997 45 96

[14] Gepner, Doron Fusion rings and geometry Comm. Math. Phys. 1991 381 411

[15] Givental, Alexander, Kim, Bumsig Quantum cohomology of flag manifolds and Toda lattices Comm. Math. Phys. 1995 609 641

[16] Griffiths, Phillip, Harris, Joseph Principles of algebraic geometry 1994

[17] Kim, Bumsig Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings Internat. Math. Res. Notices 1995 1 15

[18] Kirillov, Anatol N. Quantum Schubert polynomials and quantum Schur functions Internat. J. Algebra Comput. 1999 385 404

[19] Kirillov, Anatol N., Maeno, Toshiaki Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula Discrete Math. 2000 191 223

[20] Kostant, Bertram Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight 𝜌 Selecta Math. (N.S.) 1996 43 91

[21] Kostant, Bertram Quantum cohomology of the flag manifold as an algebra of rational functions on a unipotent algebraic group 1997 157 175

[22] Lascoux, Alain, Schützenberger, Marcel-Paul Polynômes de Schubert C. R. Acad. Sci. Paris Sér. I Math. 1982 447 450

[23] Lusztig, G. Total positivity in reductive groups 1994 531 568

[24] Lusztig, George Total positivity and canonical bases 1997 281 295

[25] Lusztig, G. Total positivity in partial flag manifolds Represent. Theory 1998 70 78

[26] Macdonald, I. G. Symmetric functions and Hall polynomials 1995

[27] Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces 1999

[28] Mcduff, Dusa, Salamon, Dietmar 𝐽-holomorphic curves and quantum cohomology 1994

[29] Minc, Henryk Nonnegative matrices 1988

[30] Okun′Kov, A. On representations of the infinite symmetric group Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 1997

[31] Siebert, Bernd, Tian, Gang On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator Asian J. Math. 1997 679 695

[32] Witten, Edward The Verlinde algebra and the cohomology of the Grassmannian 1995 357 422