Grassmann–Grassmann conormal varieties, integrability, and plane partitions
Annales de l'Institut Fourier, to appear, 59 p.

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 : 2017-04-12
Revised : 2017-10-16
Accepted : 2018-02-02
Classification:  14M15,  82B23,  14Q99
Keywords: Quantum Knizhnik–Zamolodchikov equation, equivariant K-theory, cotangent bundle of the Grassmannian, loop model
@unpublished{AIF_0__0_0_A30_0,
     author = {Knutson, Allen and Zinn-Justin, Paul},
     title = {Grassmann--Grassmann conormal varieties, integrability, and plane partitions},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Knutson, Allen; Zinn-Justin, Paul. Grassmann–Grassmann conormal varieties, integrability, and plane partitions. Annales de l'Institut Fourier, to appear, 59 p.

[1] Brion, Michel Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Birkhäuser (Trends in Mathematics) (2005), pp. 33-85 https://www-fourier.ujf-grenoble.fr/~mbrion/lecturesrev.pdf | Article | MR 2143072

[2] Cantini, Luigi; Sportiello, Andrea Proof of the Razumov–Stroganov conjecture, J. Comb. Theory, Ser. A, Tome 118 (2011) no. 5, pp. 1549-1574 | Article | MR 2771600 | Zbl 1232.05038

[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 MR2278472

[4] Di Francesco, Philippe; Zinn-Justin, Paul Around the Razumov–Stroganov conjecture: proof of a multi-parameter sum rule, Electron. J. Comb., Tome 12 (2005), 6, 27 pages http://www.combinatorics.org/volume_12/abstracts/v12i1r6.html (Art. ID 6, 27 pages) | MR MR2134169 | Zbl 1108.05013

[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., Tome 38 (2005) no. 48, p. L815-L822 | Article | MR MR2185933 | Zbl 1078.81037

[6] Di Francesco, Philippe; Zinn-Justin, Paul Quantum Knizhnik–Zamolodchikov equation, Totally Symmetric Self-Complementary Plane Partitions and Alternating Sign Matrices, Theor. Math. Phys., Tome 154 (2008) no. 3, pp. 331-348 | Article | Zbl 1192.81308

[7] Di Francesco, Philippe; Zinn-Justin, Paul; Zuber, Jean-Bernard A bijection between classes of fully packed loops and plane partitions, Electron. J. Comb., Tome 11 (2004) no. 1, 64, 11 pages (Art. ID 64, 11 pages) | MR MR2097330 | Zbl 1054.05010

[8] Eisenbud, David Commutative algebra. With a view toward algebraic geometry, Springer, Graduate Texts in Mathematics, Tome 150 (1995), xvi+785 pages | Article | MR 1322960 | Zbl 0819.13001

[9] Eisenbud, David; Sturmfels, Bernd Binomial ideals, Duke Math. J., Tome 84 (1996) no. 1, pp. 1-45 | Article | MR 1394747 | Zbl 0873.13021

[10] Fulton, William Introduction to toric varieties, Princeton University Press, Annals of Mathematics Studies, Tome 131 (1993), xii+157 pages (The William H. Roever Lectures in Geometry) | Article | MR 1234037 | Zbl 0813.14039

[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., Tome 2005 (2005) no. 28, pp. 1717-1742 | Article | MR 2172339 | Zbl 1122.20004

[13] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. Math., Tome 161 (2005) no. 3, pp. 1245-1318 | MR MR2180402 | Zbl 1089.14007

[14] Knutson, Allen; Miller, Ezra; Shimozono, Mark Four positive formulae for type A quiver polynomials, Invent. Math., Tome 166 (2006) no. 2, pp. 229-325 | Article | MR 2249801 | Zbl 1107.14046

[15] Knutson, Allen; Miller, Ezra; Yong, Alexander Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., Tome 630 (2009), pp. 1-31 | Article | MR 2526784 | Zbl 1169.14033

[16] Knutson, Allen; Zinn-Justin, Paul The Brauer loop scheme and orbital varieties, J. Geom. Phys., Tome 78 (2014), pp. 80-110 | Article | Zbl 1326.81094

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

[18] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Springer, Graduate Texts in Mathematics, Tome 227 (2005), xiv+417 pages | MR 2110098 | Zbl 1066.13001

[19] Pasquier, Vincent Quantum incompressibility and Razumov Stroganov type conjectures, Ann. Henri Poincaré, Tome 7 (2006) no. 3, pp. 397-421 | MR MR2226742 | Zbl 1098.81098

[20] Razumov, Alexander; Stroganov, Yu. Combinatorial nature of the ground-state vector of the O(1) loop model, Teor. Mat. Fiz., Tome 138 (2004) no. 3, pp. 395-400 | Article | MR MR2077318 | Zbl 1178.82020

[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 )

[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., Tome 62 (2012) no. 11, pp. 2188-2207 | Article | MR 2964653 | Zbl 1254.22013

[23] Rosu, Ioanid Equivariant K-theory and equivariant cohomology, Math. Z., Tome 243 (2003) no. 3, pp. 423-448 (With an appendix by Allen Knutson and Rosu) | Article | MR 1970011 | Zbl 1019.19003

[24] Rothbach, Brian Borel orbits of X 2 =0 in 𝔤𝔩 n , University of California (USA) (2009) (Ph. D. Thesis)

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

[26] Vezzosi, Gabriele; Vistoli, Angelo Higher algebraic K-theory of group actions with finite stabilizers, Duke Math. J., Tome 113 (2002) no. 1, pp. 1-55 | Article | MR 1905391 | Zbl 1012.19002

[27] Weyman, Jerzy Cohomology of vector bundles and syzygies, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 149 (2003), xiv+371 pages | Article | MR 1988690 | Zbl 1075.13007

[28] Wieland, Benjamin A large dihedral symmetry of the set of alternating sign matrices, Electron. J. Comb., Tome 7 (2000), 37, 13 pages (Art. ID 37, 13 pages) | MR MR1773294 | Zbl 0956.05015

[29] Woo, Alexander; Yong, Alexander When is a Schubert variety Gorenstein?, Adv. Math., Tome 207 (2006) no. 1, pp. 205-220 | Article | MR 2264071 | Zbl 1112.14058

[30] Zinn-Justin, Paul Proof of the Razumov–Stroganov conjecture for some infinite families of link patterns, Electron. J. Comb., Tome 13 (2006) no. 1, 110, 15 pages (Art. ID 110, 15 pages) | MR MR2274325 | Zbl 1119.82018

[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., Tome 185 (2015) no. 3, pp. 1741-1758 | Article | Zbl 1338.81229

[33] Zuber, Jean-Bernard On the counting of Fully Packed Loop configurations: some new conjectures, Electron. J. Comb., Tome 11 (2004) no. 1, 13, 15 pages http://www.combinatorics.org/volume_11/abstracts/v11i1r13.html (Art. ID 13, 15 pages) | MR MR2035307 | Zbl 1054.05011