no. 3
Grassmann–Grassmann conormal varieties, integrability, and plane partitions
Knutson, Allen1; Zinn-Justin, Paul2
1 Cornell Ithaca, NY 14853 (USA)
2 School of Mathematics and Statistics The University of Melbourne Parkville, Victoria 3010 (Australia)
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1087-1145.

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant K-class is given by the partition function of an integrable loop model, and furthermore their K-theoretic pushforward to a point is a solution of the level 1 quantum Knizhnik–Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov–Stroganov correspondence satisfied by the loop model.

Nous donnons une formule conjecturelle pour des faisceaux coherents de support des variétés conormales dans le fibré cotangent de la Grassmannienne, tels que leur classe de K-théorie équivariante est donnée par la fonction de partition d’un modèle de boucles intégrable, et que de plus leur image dans la K-théorie d’un point est solution de l’équation de Knizhnik–Zamolodchikov quantique de niveau 1. Nous démontrons ces résultats dans le cas où la Lagrangienne est lisse (donc le fibré conormal d’une sous-Grassmannienne). Pour pousser en avant vers un point, ou de manière équivalente vers son affinisation, nous dégénérons simultanément la Lagrangienne et son faisceau (sur l’affinisation) ; le faisceau dégénère en une somme directe de modules cycliques sur les composantes géométriques, qui sont en bijection avec des partitions planes, ce qui donne une interprétation géométrique à la correspondance de Razumov–Stroganov satisfaite par le modèle de boucles.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3266
Classification: 14M15, 82B23, 14Q99
Keywords: Quantum Knizhnik–Zamolodchikov equation, equivariant $K$-theory, cotangent bundle of the Grassmannian, loop model
Mots-clés : Equation de Knizhnik–Zamolodchikov quantique, $K$-théorie équivariante, fibré cotangent de la Grassmannienne, modèle de boucles

Knutson, Allen 1; Zinn-Justin, Paul 2

1 Cornell Ithaca, NY 14853 (USA)
2 School of Mathematics and Statistics The University of Melbourne Parkville, Victoria 3010 (Australia)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Knutson, Allen; Zinn-Justin, Paul. Grassmann–Grassmann conormal varieties, integrability, and plane partitions. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1087-1145. doi : 10.5802/aif.3266. https://aif.centre-mersenne.org/articles/10.5802/aif.3266/

[1] Brion, Michel Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties (Trends in Mathematics), Birkhäuser, 2005, pp. 33-85 https://www-fourier.ujf-grenoble.fr/~mbrion/lecturesrev.pdf | DOI | MR

[2] Cantini, Luigi; Sportiello, Andrea Proof of the Razumov–Stroganov conjecture, J. Comb. Theory, Ser. A, Volume 118 (2011) no. 5, pp. 1549-1574 | DOI | MR | Zbl

[3] Di Francesco, Philippe Totally symmetric self-complementary plane partitions and the quantum Knizhnik–Zamolodchikov equation: a conjecture, J. Stat. Mech. Theory Exp. (2006) no. 9, P09008, 14 pages (Art. ID P09008, 14 pages) | MR

[4] Di Francesco, Philippe; Zinn-Justin, Paul Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule, Electron. J. Comb., Volume 12 (2005), 6, 27 pages http://www.combinatorics.org/volume_12/abstracts/v12i1r6.html (Art. ID 6, 27 pages) | MR | Zbl

[5] Di Francesco, Philippe; Zinn-Justin, Paul Quantum Knizhnik–Zamolodchikov equation, generalized Razumov–Stroganov sum rules and extended Joseph polynomials, J. Phys. A, Math. Gen., Volume 38 (2005) no. 48, p. L815-L822 | DOI | MR | Zbl

[6] Di Francesco, Philippe; Zinn-Justin, Paul Quantum Knizhnik–Zamolodchikov equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices, Theor. Math. Phys., Volume 154 (2008) no. 3, pp. 331-348 | DOI | Zbl

[7] Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard A bijection between classes of fully packed loops and plane partitions, Electron. J. Comb., Volume 11 (2004) no. 1, 64, 11 pages (Art. ID 64, 11 pages) | MR | Zbl

[8] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995, xvi+785 pages | DOI | MR | Zbl

[9] Eisenbud, David; Sturmfels, Bernd Binomial ideals, Duke Math. J., Volume 84 (1996) no. 1, pp. 1-45 | DOI | MR | Zbl

[10] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, 1993, xii+157 pages (The William H. Roever Lectures in Geometry) | DOI | MR | Zbl

[11] Grayson, Daniel; Stillman, Michael Macaulay2, a software system for research in algebraic geometry (Available at http://www.math.uiuc.edu/Macaulay2/)

[12] Kasatani, Masahiro Subrepresentations in the polynomial representation of the double affine Hecke algebra of type GL n at t k+1 q r-1 =1, Int. Math. Res. Not., Volume 2005 (2005) no. 28, pp. 1717-1742 | DOI | MR | Zbl

[13] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | MR | Zbl

[14] Knutson, Allen; Miller, Ezra; Shimozono, Mark Four positive formulae for type A quiver polynomials, Invent. Math., Volume 166 (2006) no. 2, pp. 229-325 | DOI | MR | Zbl

[15] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Volume 630 (2009), pp. 1-31 | DOI | MR | Zbl

[16] Knutson, Allen; Zinn-Justin, Paul The Brauer loop scheme and orbital varieties, J. Geom. Phys., Volume 78 (2014), pp. 80-110 | DOI | MR | Zbl

[17] Maulik, Davesh; Okounkov, Andrei Quantum groups and quantum cohomology (2012) (https://arxiv.org/abs/1211.1287) | Zbl

[18] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer, 2005, xiv+417 pages | MR | Zbl

[19] Pasquier, Vincent Quantum incompressibility and Razumov Stroganov type conjectures, Ann. Henri Poincaré, Volume 7 (2006) no. 3, pp. 397-421 | DOI | MR | Zbl

[20] Razumov, Alexander; Stroganov, Yu. Combinatorial nature of the ground-state vector of the O(1) loop model, Teor. Mat. Fiz., Volume 138 (2004) no. 3, pp. 395-400 | DOI | MR | Zbl

[21] Rimányi, Richárd; Tarasov, Vitaly; Varchenko, Alexander Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety (2014) (https://arxiv.org/abs/1411.0478) | Zbl

[22] Rimányi, Richárd; Tarasov, Vitaly; Varchenko, Alexander; Zinn-Justin, Paul Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral, J. Geom. Phys., Volume 62 (2012) no. 11, pp. 2188-2207 | DOI | MR | Zbl

[23] Rosu, Ioanid Equivariant K-theory and equivariant cohomology, Math. Z., Volume 243 (2003) no. 3, pp. 423-448 (With an appendix by Allen Knutson and Rosu) | DOI | MR | Zbl

[24] Rothbach, Brian Borel orbits of X 2 =0 in 𝔤𝔩 n , Ph. D. Thesis, University of California (USA) (2009) (http://search.proquest.com//docview/304845738)

[25] Su, Changjian Restriction formula for stable basis of Springer resolution (2015) (https://arxiv.org/abs/1501.04214) | Zbl

[26] Vezzosi, Gabriele; Vistoli, Angelo Higher algebraic K-theory of group actions with finite stabilizers, Duke Math. J., Volume 113 (2002) no. 1, pp. 1-55 | DOI | MR | Zbl

[27] Weyman, Jerzy Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, 149, Cambridge University Press, 2003, xiv+371 pages | DOI | MR | Zbl

[28] Wieland, Benjamin A large dihedral symmetry of the set of alternating sign matrices, Electron. J. Comb., Volume 7 (2000), 37, 13 pages (Art. ID 37, 13 pages) | MR | Zbl

[29] Woo, Alexander; Yong, Alexander When is a Schubert variety Gorenstein?, Adv. Math., Volume 207 (2006) no. 1, pp. 205-220 | DOI | MR | Zbl

[30] Zinn-Justin, Paul Proof of the Razumov–Stroganov conjecture for some infinite families of link patterns, Electron. J. Comb., Volume 13 (2006) no. 1, 110, 15 pages (Art. ID 110, 15 pages) | MR | Zbl

[31] Zinn-Justin, Paul Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009 http://www.lpthe.jussieu.fr/~pzinn/publi/hdr.pdf (Habilitation thesis)

[32] Zinn-Justin, Paul Quiver varieties and the quantum Knizhnik–Zamolodchikov equation, Theor. Math. Phys., Volume 185 (2015) no. 3, pp. 1741-1758 | DOI | MR | Zbl

[33] Zuber, Jean-Bernard On the counting of Fully Packed Loop configurations: some new conjectures, Electron. J. Comb., Volume 11 (2004) no. 1, 13, 15 pages http://www.combinatorics.org/volume_11/abstracts/v11i1r13.html (Art. ID 13, 15 pages) | MR | Zbl

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