Algebraic weaves and braid varieties.(English)Zbl 07958212
Summary: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting decompositions of braid varieties and their quotients. It is shown that the maximal charts of these decompositions are exponential Darboux charts for the holomorphic symplectic structures, and we relate these charts to exact Lagrangian fillings of Legendrian links.
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References:
| [1] | V. I. Arnold, Lagrange and Legendre cobordisms. I, Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, 1-13. ·Zbl 0448.57017 |
| [2] | Singularities of Caustics and Wave Fronts, Math. Appl. (Soviet Ser.), vol. 62, Kluwer Aca-demic Publishers Group, Dordrecht, 1990. ·Zbl 0734.53001 |
| [3] | M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523-615. ·Zbl 0509.14014 |
| [4] | D. Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87-161. ·Zbl 0573.58022 |
| [5] | M. V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Inst. Hautes Études Sci. Publ. Math. (1988), no. 68, 211-221 (1989). ·Zbl 0701.58012 |
| [6] | P. Boalch, Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. Math. 146 (2001), no. 3, 479-506. ·Zbl 1044.53060 |
| [7] | Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), no. 2, 137-205. ·Zbl 1001.53059 |
| [8] | Quasi-Hamiltonian geometry of meromorphic connections, Duke Math. J. 139 (2007), no. 2, 369-405. ·Zbl 1126.53055 |
| [9] | Wild character varieties, points on the Riemann sphere and Calabi’s examples, Represen-tation Theory, Special Functions and Painlevé Equations-RIMS 2015, Adv. Stud. Pure Math., vol. 76, Math. Soc. Japan, Tokyo, 2018, pp. 67-94. ·Zbl 1409.53068 |
| [10] | F. Bourgeois and B. Chantraine, Bilinearized Legendrian contact homology and the augmentation cate-gory, J. Symplectic Geom. 12 (2014), no. 3, 553-583. ·Zbl 1308.53119 |
| [11] | F. Bourgeois, J. M. Sabloff, and L. Traynor, Lagrangian cobordisms via generating families: construction and geography, Algebr. Geom. Topol. 15 (2015), no. 4, 2439-2477. ·Zbl 1330.57037 |
| [12] | S. B. Brodsky and C. Stump, Towards a uniform subword complex description of acyclic finite type cluster algebras, Algebr. Comb. 1 (2018), no. 4, 545-572. ·Zbl 1423.13118 |
| [13] | M. Broué and J. Michel, Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser, Boston, MA, 1997, pp. 73-139. ·Zbl 1029.20500 |
| [14] | T. Brüstle, G. Dupont, and M. Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN 2014 (2014), no. 16, 4547-4586. ·Zbl 1346.16009 |
| [15] | T. Brüstle and D. Yang, Ordered exchange graphs, Advances in Representation Theory of Algebras, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2013, pp. 135-193. |
| [16] | R. Casals, Lagrangian skeleta and plane curve singularities, J. Fixed Point Theory Appl. 24 (2022), no. 2, Paper No. 34. ·Zbl 1501.53088 |
| [17] | R. Casals and H. Gao, Infinitely many Lagrangian fillings, Ann. of Math. (2) 195 (2022), no. 1, 207-249. [141.66.176.121] Project MUSE (2024-12-04 12:04 GMT) FIZ Karlsruhe -Leibniz Institute for Information Infrastructure [18] R. Casals, E. Gorsky, M. Gorsky, I. Le, L. Shen, and J. Simental, Cluster structures on braid varieties, J. Amer. Math. Soc. (to appear). |
| [18] | R. Casals, E. Gorsky, M. Gorsky, and J. Simental, Positroid links and Braid varieties, preprint, https://arxiv.org/abs/2105.13948. |
| [19] | R. Casals, I. Le, M. Sherman-Bennett, and D. Weng, Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas), preprint, https://arxiv.org/abs/ 2308.06184. |
| [20] | R. Casals and E. Murphy, Differential algebra of cubic planar graphs, Adv. Math. 338 (2018), 401-446. ·Zbl 1397.05042 |
| [21] | Legendrian fronts for affine varieties, Duke Math. J. 168 (2019), no. 2, 225-323. ·Zbl 1490.57034 |
| [22] | R. Casals and L. Ng, Braid loops with infinite monodromy on the Legendrian contact DGA, J. Topol. 15 (2022), no. 4, 1927-2016. ·Zbl 1527.53080 |
| [23] | R. Casals and D. Weng, Microlocal theory of Legendrian links and cluster algebras, Geom. Topol. 28 (2024), no. 2, 901-1000. ·Zbl 1547.13030 |
| [24] | R. Casals and E. Zaslow, Legendrian weaves: N -graph calculus, flag moduli and applications, Geom. Topol. 26 (2022), no. 8, 3589-3745. ·Zbl 1521.53061 |
| [25] | C. Ceballos, J.-P. Labbé, and C. Stump, Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin. 39 (2014), no. 1, 17-51. ·Zbl 1286.05180 |
| [26] | J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no. 39, 5-173. ·Zbl 0213.25202 |
| [27] | Y. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), no. 3, 441-483. ·Zbl 1029.57011 |
| [28] | J. F. Davis, P. Hersh, and E. Miller, Fibers of maps to totally nonnegative spaces, preprint, https://arxiv.org/ abs/1903.01420. |
| [29] | P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math. 128 (1997), no. 1, 159-175. ·Zbl 0879.57017 |
| [30] | M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 53-88. ·Zbl 0312.14009 |
| [31] | V. V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499-511. ·Zbl 0563.14023 |
| [32] | O. Dudas, Note on the Deodhar decomposition of a double Schubert cell, preprint, https://arxiv.org/abs/ 0807.2198. |
| [33] | T. Ekholm, K. Honda, and T. Kálmán, Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 11, 2627-2689. ·Zbl 1357.57044 |
| [34] | B. Elias, Thicker Soergel calculus in type A, Proc. Lond. Math. Soc. (3) 112 (2016), no. 5, 924-978. ·Zbl 1388.20011 |
| [35] | A diamond lemma for Hecke-type algebras, Trans. Amer. Math. Soc. 375 (2022), no. 3, 1883-1915. ·Zbl 1497.20044 |
| [36] | B. Elias and M. Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. (2010), Paper No. 978635. ·Zbl 1219.18003 |
| [37] | B. Elias and G. Williamson, Soergel calculus, Represent. Theory 20 (2016), 295-374. ·Zbl 1427.20006 |
| [38] | L. Escobar, Brick manifolds and toric varieties of brick polytopes, Electron. J. Combin. 23 (2016), no. 2, Paper No. 2.25. ·Zbl 1368.14064 |
| [39] | J. B. Etnyre and L. L. Ng, Legendrian contact homology in R 3 , Surveys in Differential Geometry 2020. Surveys in 3-Manifold Topology and Geometry, Surv. Differ. Geom., vol. 25, Int. Press, Boston, MA, 2022, pp. 103-161. ·Zbl 1517.57010 |
| [40] | L. Euler, Specimen algorithmi singularis, Novi Commentarii academiae scientiarum Petropolitanae 9 (1764), 53-69. |
| [41] | S. Felsner and H. Weil, A theorem on higher Bruhat orders, Discrete Comput. Geom. 23 (2000), no. 1, 121-127. ·Zbl 0996.52016 |
| [42] | S. Fomin and C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), no. 1-3, 179-200. ·Zbl 1011.05062 |
| [43] | D. Fuchs, Chekanov-Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003), no. 1, 43-65. ·Zbl 1028.57005 |
| [44] | P. Galashin and T. Lam, Positroids, knots, and q, t-Catalan numbers, Duke Math. J. (to appear). ·Zbl 1503.14044 |
| [45] | P. Galashin, T. Lam, and M. Sherman-Bennett, Braid variety cluster structures, II: general type, preprint, https://arxiv.org/abs/2301.07268. |
| [46] | P. Galashin, T. Lam, M. Sherman-Bennett, and D. Speyer, Braid variety cluster structures, I: 3D plabic graphs, preprint, https://arxiv.org/abs/2210.04778. |
| [47] | H. Gao, L. Shen, and D. Weng, Augmentations, fillings, and clusters, Geom. Funct. Anal. 34 (2024), no. 3, 798-867. ·Zbl 07862380 |
| [48] | H. Geiges, An introduction to contact topology, Cambridge Stud. Adv. Math., vol. 109, Cambridge Uni-versity Press, Cambridge, 2008. ·Zbl 1153.53002 |
| [49] | W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200-225. ·Zbl 0574.32032 |
| [50] | A. B. Goncharov and R. Kenyon, Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 747-813. ·Zbl 1288.37025 |
| [51] | E. Gorsky, M. Hogancamp, and A. Mellit, Tautological classes and symmetry in Khovanov-Rozansky homology, preprint, https://arxiv.org/abs/2103.01212. |
| [52] | M. A. Gorsky, Subword complexes and nil-Hecke moves, Model. Anal. Inf. Sistem 20 (2013), no. 6, 121-128. |
| [53] | Subword complexes and 2-truncated cubes, Russ. Math. Surv. 69 (2014), no. 3, 572-574. ·Zbl 1316.05131 |
| [54] | Subword complexes and edge subdivisions, Proc. Steklov Inst. Math. 286 (2014), no. 1, 114-127. ·Zbl 1338.20037 |
| [55] | S. Guillermou, M. Kashiwara, and P. Schapira, Sheaf quantization of Hamiltonian isotopies and applica-tions to nondisplaceability problems, Duke Math. J. 161 (2012), no. 2, 201-245. ·Zbl 1242.53108 |
| [56] | M. B. Henry and D. Rutherford, Equivalence classes of augmentations and Morse complex sequences of Legendrian knots, Algebr. Geom. Topol. 15 (2015), no. 6, 3323-3353. ·Zbl 1334.57025 |
| [57] | Ruling polynomials and augmentations over finite fields, J. Topol. 8 (2015), no. 1, 1-37. ·Zbl 1312.57033 |
| [58] | P. Hersh, Regular cell complexes in total positivity, Invent. Math. 197 (2014), no. 1, 57-114. ·Zbl 1339.20041 |
| [59] | F. Hivert, A. Schilling, and N. M. Thiéry, Hecke group algebras as quotients of affine Hecke algebras at level 0, J. Combin. Theory Ser. A 116 (2009), no. 4, 844-863. ·Zbl 1185.20004 |
| [60] | L. Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79-183. ·Zbl 0212.46601 |
| [61] | Linear differential operators, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars Éditeur, Paris, 1971, pp. 121-133. ·Zbl 0223.35083 |
| [62] | J. Hughes, Lagrangian fillings in A-type and their Kálmán loop orbits, Rev. Mat. Iberoam. 39 (2023), no. 5, 1681-1723. ·Zbl 1543.53072 |
| [63] | D. Jahn, R. Löwe, and C. Stump, Minkowski decompositions for generalized associahedra of acyclic type, Algebr. Comb. 4 (2021), no. 5, 757-775. ·Zbl 1481.13037 |
| [64] | L. C. Jeffrey, Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, Duke Math. J. 77 (1995), no. 2, 407-429. ·Zbl 0870.57013 |
| [65] | T. Kálmán, Contact homology and one parameter families of Legendrian knots, Geom. Topol. 9 (2005), 2013-2078. ·Zbl 1095.53059 |
| [66] | Braid-positive Legendrian links, Int. Math. Res. Not. IMRN 2006 (2006), Paper No. 14874. ·Zbl 1128.57006 |
| [67] | Meridian twisting of closed braids and the Homfly polynomial, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 649-660. ·Zbl 1183.57006 |
| [68] | M. Kashiwara and P. Schapira, Micro-support des faisceaux: application aux modules différentiels, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 8, 487-490. ·Zbl 0501.58006 |
| [69] | Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer-Verlag, Berlin, 1990. ·Zbl 0709.18001 |
| [70] | L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge Theory to Integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 87-174. ·Zbl 1206.14009 |
| [71] | N. Kitchloo, Symmetry breaking and link homologies i, preprint, https://arxiv.org/abs/1910.07443. |
| [72] | A. Knutson, T. Lam, and D. E. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710-1752. ·Zbl 1330.14086 |
| [73] | A. Knutson and E. Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), no. 1, 161-176. ·Zbl 1069.20026 |
| [74] | Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245-1318. ·Zbl 1089.14007 |
| [75] | T. Lam and D. E. Speyer, Cohomology of cluster varieties I: Locally acyclic case, Algebra Number Theory 16 (2022), no. 1, 179-230. ·Zbl 1498.13062 |
| [76] | B. Leclerc, Cluster structures on strata of flag varieties, Adv. Math. 300 (2016), 190-228. ·Zbl 1375.13036 |
| [77] | B. Leclerc and A. Zelevinsky, Quasicommuting families of quantum Plücker coordinates, Kirillov’s Sem-inar on Representation Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 85-108. ·Zbl 0894.14021 |
| [78] | Yu. I. Manin, Correspondences, motifs and monoidal transformations, Math. USSR Sb. 6 (1968), no. 4, 439-470. |
| [79] | Higher Bruhat orders, related to the symmetric group, Funct. Anal. Appl. 20 (1986), 148-150. ·Zbl 0646.20014 |
| [80] | Yu. I. Manin and V. V. Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, Algebraic Number Theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 289-308. ·Zbl 0759.20002 |
| [81] | T. McConville, Homotopy type of intervals of the second higher Bruhat orders, Order 35 (2018), no. 3, 515-524. ·Zbl 1408.52037 |
| [82] | A. Mellit, private communication. |
| [83] | Cell decompositions of character varieties, preprint, https://arxiv.org/abs/1905.10685. |
| [84] | L. Ng and D. Rutherford, Satellites of Legendrian knots and representations of the Chekanov-Eliashberg algebra, Algebr. Geom. Topol. 13 (2013), no. 5, 3047-3097. ·Zbl 1280.57019 |
| [85] | L. Ng, D. Rutherford, V. Shende, S. Sivek, and E. Zaslow, Augmentations are sheaves, Geom. Topol. 24 (2020), no. 5, 2149-2286. ·Zbl 1457.53064 |
| [86] | P. N. Norton, 0-Hecke algebras, J. Austral. Math. Soc. Ser. A 27 (1979), no. 3, 337-357. ·Zbl 0407.16019 |
| [87] | U. Pachner, P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), no. 2, 129-145. ·Zbl 0729.52003 |
| [88] | Y. Pan, Exact Lagrangian fillings of Legendrian (2, n) torus links, Pacific J. Math. 289 (2017), no. 2, 417-441. ·Zbl 1432.53127 |
| [89] | Y. Pan and D. Rutherford, Functorial LCH for immersed Lagrangian cobordisms, J. Symplectic Geom. 19 (2021), no. 3, 635-722. ·Zbl 1478.57036 |
| [90] | Augmentations and immersed Lagrangian fillings, J. Topol. 16 (2023), no. 1, 368-429. ·Zbl 1543.53082 |
| [91] | V. Pilaud and C. Stump, Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math. 276 (2015), 1-61. ·Zbl 1405.05196 |
| [92] | N. Reading, From the Tamari lattice to Cambrian lattices and beyond, Associahedra, Tamari lattices and related structures, Progr. Math., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 293-322. ·Zbl 1292.20044 |
| [93] | R. W. Richardson and T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1-3, 389-436. ·Zbl 0704.20039 |
| [94] | Combinatorics and geometry of K-orbits on the flag manifold, Linear algebraic groups and their representations (Los Angeles, CA, 1992), Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109-142. ·Zbl 0840.20039 |
| [95] | R. Rouquier, Categorification of sl 2 and braid groups, Trends in representation theory of algebras and related topics, Contemp. Math., vol. 406, Amer. Math. Soc., Providence, RI, 2006, pp. 137-167. ·Zbl 1162.20301 |
| [96] | T. Scroggin, On the cohomology of two stranded braid varieties, preprint, https://arxiv.org/abs/ 2312.03283. |
| [97] | K. Serhiyenko and M. Sherman-Bennett, Leclerc’s conjecture on a cluster structure for type A Richardson varieties, Adv. Math. 447 (2024), Paper No. 109698. ·Zbl 1553.13024 |
| [98] | K. Serhiyenko, M. Sherman-Bennett, and L. Williams, Cluster structures in Schubert varieties in the Grassmannian, Proc. Lond. Math. Soc. (3) 119 (2019), no. 6, 1694-1744. ·Zbl 1439.05227 |
| [99] | L. Shen and D. Weng, Cluster structures on double Bott-Samelson cells, Forum Math. Sigma 9 (2021), Paper No. e66. ·Zbl 1479.13028 |
| [100] | V. Shende, D. Treumann, H. Williams, and E. Zaslow, Cluster varieties from Legendrian knots, Duke Math. J. 168 (2019), no. 15, 2801-2871. ·Zbl 1475.53094 |
| [101] | V. Shende, D. Treumann, and E. Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017), no. 3, 1031-1133. ·Zbl 1369.57016 |
| [102] | D. E. Speyer, Richardson varieties, projected Richardson varieties and positroid varieties, preprint, https://arxiv.org/abs/2303.04831. |
| [103] | G. G. Stokes, On the numerical calculation of a class of definite integrals and infinite series, Trans. Camb. Phil. Soc. 9 (1847), 379-407. |
| [104] | H. Thomas, Maps between higher Bruhat orders and higher Stasheff-Tamari posets, Formal Power Series and Algebraic Combinatorics Conference (Linköping University, Sweden), 2003. |
| [105] | S. V. Tsaranov, Representation and classification of Coxeter monoids, European J. Combin. 11 (1990), no. 2, 189-204. ·Zbl 0703.20055 |
| [106] | G. M. Ziegler, Higher Bruhat orders and cyclic hyperplane arrangements, Topology 32 (1993), no. 2, 259-279 ·Zbl 0782.06003 |
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