Geodesic
Loop Models and $K$-Theory
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems ($R$-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for $K$-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties.
Keywords: quantum integrability; loop models; $K$-theory. 
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     author = {Paul Zinn-Justin},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a68/}
}
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Paul Zinn-Justin. Loop Models and $K$-Theory. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a68/

[1] Aganagic M., Okounkov A., Elliptic stable envelope, arXiv: 1604.00423

[2] Bender M., Perrin N., Singularities of closures of spherical $B$-conjugacy classes of nilpotent orbits,, arXiv: 1412.5654

[3] Birman J. S., Wenzl H., “Braids, link polynomials and a new algebra”, Trans. Amer. Math. Soc., 313 (1989), 249–273 | DOI | MR | Zbl

[4] Brauer R., “On algebras which are connected with the semisimple continuous groups”, Ann. of Math., 38 (1937), 857–872 | DOI | MR

[5] Brion M., “Lectures on the geometry of flag varieties”, Topics in Cohomological Studies of Algebraic Varieties, Trends Math., Birkhäuser, Basel, 2005, 33–85, arXiv: math.AG/0410240 | DOI

[6] Cantini L., Sportiello A., “Proof of the Razumov–Stroganov conjecture”, J. Combin. Theory Ser. A, 118 (2011), 1549–1574, arXiv: 1003.3376 | DOI | MR | Zbl

[7] Di Francesco P., Zinn-Justin P., “The quantum Knizhnik–Zamolodchikov equation, generalized Razumov–Stroganov sum rules and extended Joseph polynomials”, J. Phys. A: Math. Gen., 38 (2005), L815–L822, arXiv: math-ph/0508059 | DOI | Zbl

[8] Fomin S., Kirillov A. N., Yang–Baxter equation, symmetric functions and Grothendieck polynomials, arXiv: hep-th/9306005

[9] Fomin S., Kirillov A. N., “The Yang–Baxter equation, symmetric functions, and Schubert polynomials”, Discrete Math., 153 (1996), 123–143 | DOI | MR | Zbl

[10] Fonseca T., Zinn-Justin P., “On some ground state components of the $O(1)$ loop model”, J. Stat. Mech. Theory Exp., 2009 (2009), P03025, 29 pp., arXiv: 0901.1679 | DOI

[11] Frenkel I. B., Reshetikhin N. Yu., “Quantum affine algebras and holonomic difference equations”, Comm. Math. Phys., 146 (1992), 1–60 | DOI | MR | Zbl

[12] Ginzburg V., Kapranov M., Vasserot E., Elliptic algebras and equivariant elliptic cohomology, arXiv: q-alg/9505012

[13] Gorbounov V., Korff C., “Quantum integrability and generalised quantum Schubert calculus”, Adv. Math., 313 (2017), 282–356, arXiv: 1408.4718 | DOI | MR | Zbl

[14] Gorbounov V., Rimányi R., Tarasov V., Varchenko A., “Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra”, J. Geom. Phys., 74 (2013), 56–86, arXiv: 1204.5138 | DOI | MR | Zbl

[15] Grayson D., Stillman M., Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/

[16] Knutson A., Miller E., “Subword complexes in Coxeter groups”, Adv. Math., 184 (2004), 161–176, arXiv: math.CO/0309259 | DOI | MR | Zbl

[17] Knutson A., Miller E., “Gröbner geometry of Schubert polynomials”, Ann. of Math., 161 (2005), 1245–1318 | DOI | MR | Zbl

[18] Knutson A., Zinn-Justin P., “A scheme related to the Brauer loop model”, Adv. Math., 214 (2007), 40–77, arXiv: math.AG/0503224 | DOI | MR | Zbl

[19] Knutson A., Zinn-Justin P., “The Brauer loop scheme and orbital varieties”, J. Geom. Phys., 78 (2014), 80–110, arXiv: 1001.3335 | DOI | MR | Zbl

[20] Knutson A., Zinn-Justin P., “Grassmann–Grassmann conormal varieties, integrability and plane partitions”, Ann. Inst. Fourier (Grenoble) (to appear) , arXiv: 1612.04465

[21] Knutson A., Zinn-Justin P., Varieties associated to loop models on finite domains, in preparation

[22] Lenart C., Zainoulline K., Zhong C., Parabolic Kazhdan–Lusztig basis, Schubert classes, and equivariant oriented cohomology, arXiv: 1608.06554

[23] Maulik D., Okounkov A., “Quantum groups and quantum cohomology”, Astérisque (to appear) , arXiv: 1211.1287

[24] Miller E., Sturmfels B., Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005 | DOI

[25] Murakami J., “The Kauffman polynomial of links and representation theory”, Osaka J. Math., 24 (1987), 745–758 | MR | Zbl

[26] Nienhuis B., “Critical and multicritical ${\rm O}(n)$ models”, Phys. A, 163 (1990), 152–157 | DOI | MR

[27] Okounkov A., Lectures on $K$-theoretic computations in enumerative geometry, arXiv: 1512.07363

[28] Ramanathan A., “Schubert varieties are arithmetically Cohen–Macaulay”, Invent. Math., 80 (1985), 283–294 | DOI | MR | Zbl

[29] Razumov A. V., Stroganov Yu. G., “Combinatorial nature of the ground-state vector of the ${\rm O}(1)$ loop model”, Theoret. and Math. Phys., 138 (2004), 333–337, arXiv: math.CO/0104216 | DOI | MR | Zbl

[30] Rimányi R., Tarasov V., Varchenko A., “Trigonometric weight functions as $K$-theoretic stable envelope maps for the cotangent bundle of a flag variety”, J. Geom. Phys., 94 (2015), 81–119, arXiv: 1411.0478 | DOI | MR | Zbl

[31] Rimányi R., Tarasov V., Varchenko A., Zinn-Justin P., “Extended Joseph polynomials, quantized conformal blocks, and a $q$-Selberg type integral”, J. Geom. Phys., 62 (2012), 2188–2207, arXiv: 1110.2187 | DOI | MR | Zbl

[32] Rothbach B. D., Borel orbits of $X^2 = 0$ in $\mathfrak{gl}_n$, Ph.D. Thesis, University of California, Berkeley, 2009 http://search.proquest.com//docview/304845738

[33] Stanley R. P., “Hilbert functions of graded algebras”, Adv. Math., 28 (1978), 57–83 | DOI | Zbl

[34] Su C., Zhao G., Zhong C., On the $K$-theory stable bases of the Springer resolution, arXiv: 1708.08013

[35] Svanes T., “Coherent cohomology on Schubert subschemes of flag schemes and applications”, Adv. Math., 14 (1974), 369–453 | DOI | MR | Zbl

[36] Woo A., Yong A., When is a Schubert variety Gorenstein?, Adv. Math., 207 (2006), 205–220, arXiv: math.AG/0409490 | DOI | MR | Zbl

[37] Zinn-Justin P., Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009, arXiv: 0901.0665

[38] Zinn-Justin P., Lectures on geometry, quantum integrability and symmetric functions, Lecture notes from a course at HSE, M., 2015 http://www.lpthe.jussieu.fr/p̃zinn/summary.pdf

[39] Zinn-Justin P., The geometry of crossing loop models, in preparation

[40] Zinn-Justin P., Di Francesco P., “Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices”, Theoret. and Math. Phys., 154 (2008), 331–348, arXiv: math-ph/0703015 | DOI | MR | Zbl