Giambelli formulae for the equivariant quantum cohomology of the Grassmannian

Mihalcea, Leonardo

doi:10.1090/S0002-9947-07-04245-6

Giambelli formulae for the equivariant quantum cohomology of the Grassmannian
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by Leonardo Constantin Mihalcea PDF

Trans. Amer. Math. Soc. 360 (2008), 2285-2301 Request permission

Abstract:

We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.

References

Alberto Arabia, Cohomologie $T$-équivariante de la variété de drapeaux d’un groupe de Kac-Moody, Bull. Soc. Math. France 117 (1989), no. 2, 129–165 (French, with English summary). MR 1015806, DOI 10.24033/bsmf.2116
Alexander Astashkevich and Vladimir Sadov, Quantum cohomology of partial flag manifolds $F_{n_1\cdots n_k}$, Comm. Math. Phys. 170 (1995), no. 3, 503–528. MR 1337131, DOI 10.1007/BF02099147
Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305. MR 1454400, DOI 10.1006/aima.1997.1627
Aaron Bertram, Ionuţ Ciocan-Fontanine, and William Fulton, Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728–746. MR 1706853, DOI 10.1006/jabr.1999.7960
L. C. Biedenharn and J. D. Louck, A new class of symmetric polynomials defined in terms of tableaux, Adv. in Appl. Math. 10 (1989), no. 4, 396–438. MR 1023942, DOI 10.1016/0196-8858(89)90023-7
Sara C. Billey, Kostant polynomials and the cohomology ring for $G/B$, Duke Math. J. 96 (1999), no. 1, 205–224. MR 1663931, DOI 10.1215/S0012-7094-99-09606-0
Michel Brion, Poincaré duality and equivariant (co)homology, Michigan Math. J. 48 (2000), 77–92. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786481, DOI 10.1307/mmj/1030132709
Anders Skovsted Buch, Quantum cohomology of Grassmannians, Compositio Math. 137 (2003), no. 2, 227–235. MR 1985005, DOI 10.1023/A:1023908007545
William Y. C. Chen and James D. Louck, The factorial Schur function, J. Math. Phys. 34 (1993), no. 9, 4144–4160. MR 1233264, DOI 10.1063/1.530032
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Alexander B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663. MR 1408320, DOI 10.1155/S1073792896000414
Alexander Givental and Bumsig Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys. 168 (1995), no. 3, 609–641. MR 1328256, DOI 10.1007/BF02101846
Ian Goulden and Curtis Greene, A new tableau representation for supersymmetric Schur functions, J. Algebra 170 (1994), no. 2, 687–703. MR 1302864, DOI 10.1006/jabr.1994.1361
Bumsig Kim, Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices 1 (1995), 1–15. MR 1317639, DOI 10.1155/S1073792895000018
Bumsig Kim, On equivariant quantum cohomology, Internat. Math. Res. Notices 17 (1996), 841–851. MR 1420551, DOI 10.1155/S1073792896000517
Bumsig Kim, Quantum cohomology of flag manifolds $G/B$ and quantum Toda lattices, Ann. of Math. (2) 149 (1999), no. 1, 129–148. MR 1680543, DOI 10.2307/121021
Allen Knutson and Terence Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), no. 2, 221–260. MR 1997946, DOI 10.1215/S0012-7094-03-11922-5
V. Lakshmibai, K. N. Raghavan, and P. Sankaran, Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Q. 2 (2006), no. 3, Special Issue: In honor of Robert D. MacPherson., 699–717. MR 2252114, DOI 10.4310/PAMQ.2006.v2.n3.a5
A. Lascoux. Interpolation - lectures at Tianjin University. June 1996.
I. G. Macdonald, Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992) Publ. Inst. Rech. Math. Av., vol. 498, Univ. Louis Pasteur, Strasbourg, 1992, pp. 5–39. MR 1308728, DOI 10.1108/EUM0000000002757
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
Leonardo Mihalcea, Equivariant quantum Schubert calculus, Adv. Math. 203 (2006), no. 1, 1–33. MR 2231042, DOI 10.1016/j.aim.2005.04.002
Leonardo Constantin Mihalcea, Positivity in equivariant quantum Schubert calculus, Amer. J. Math. 128 (2006), no. 3, 787–803. MR 2230925, DOI 10.1353/ajm.2006.0026
A. I. Molev. Factorial supersymmetric Schur functions and super Capelli identities. In Proc. of the AMS - Kirillov’s seminar on representation theory, pages 109–137, Providence, RI, 1998. Amer. Math. Soc.
Alexander I. Molev and Bruce E. Sagan, A Littlewood-Richardson rule for factorial Schur functions, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4429–4443. MR 1621694, DOI 10.1090/S0002-9947-99-02381-8
Andrei Okounkov, Quantum immanants and higher Capelli identities, Transform. Groups 1 (1996), no. 1-2, 99–126. MR 1390752, DOI 10.1007/BF02587738
A. Okounkov and O. Olshanski. Shifted Schur functions. St. Petersburg Math. J., 9(2), 1997. also available on ar$\chi$iv: math: q-alg/9605042.
Alexander Postnikov, Affine approach to quantum Schubert calculus, Duke Math. J. 128 (2005), no. 3, 473–509. MR 2145741, DOI 10.1215/S0012-7094-04-12832-5