From dimers to webs
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- by Chris Fraser, Thomas Lam and Ian Le PDF
- Trans. Amer. Math. Soc. 371 (2019), 6087-6124 Request permission
Abstract:
We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of $\mathrm {SL}_r$-webs and is built upon the $r$-fold dimer model on the network. When $r$ equals 1, our map is a reformulation of Postnikov’s boundary measurement used to coordinatize positroid strata. When $r$ equals 2 or 3, it is a reformulation of the $\mathrm {SL}_2$- and $\mathrm {SL}_3$-web immanants defined by the second author. The basic result is that the higher-rank map factors through Postnikov’s map. As an application, we deduce generators and relations for the space of $\mathrm {SL}_r$-webs, re-proving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata and thus between webs and total positivity.References
- Sabin Cautis, Joel Kamnitzer, and Scott Morrison, Webs and quantum skew Howe duality, Math. Ann. 360 (2014), no. 1-2, 351–390. MR 3263166, DOI 10.1007/s00208-013-0984-4
- Sergey Fomin and Pavlo Pylyavskyy, Webs on surfaces, rings of invariants, and clusters, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9680–9687. MR 3263299, DOI 10.1073/pnas.1313068111
- Sergey Fomin and Pavlo Pylyavskyy, Tensor diagrams and cluster algebras, Adv. Math. 300 (2016), 717–787. MR 3534844, DOI 10.1016/j.aim.2016.03.030
- K. R. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5753–5780. MR 2529913, DOI 10.1090/S0002-9947-09-04654-6
- Mikhail Khovanov and Greg Kuperberg, Web bases for $\textrm {sl}(3)$ are not dual canonical, Pacific J. Math. 188 (1999), no. 1, 129–153. MR 1680395, DOI 10.2140/pjm.1999.188.129
- Dongseok Kim, Graphical calculus on representations of quantum Lie algebras, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–University of California, Davis. MR 2704398
- Allen Knutson, Thomas Lam, and David E. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710–1752. MR 3123307, DOI 10.1112/S0010437X13007240
- Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comput. Sci. 319 (2004), no. 1-3, 29–57. MR 2074946, DOI 10.1016/j.tcs.2004.02.022
- Greg Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109–151. MR 1403861, DOI 10.1007/BF02101184
- Thomas Lam, Dimers, webs, and positroids, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 633–656. MR 3431654, DOI 10.1112/jlms/jdv039
- Thomas Lam, Totally nonnegative Grassmannian and Grassmann polytopes, Current developments in mathematics 2014, Int. Press, Somerville, MA, 2016, pp. 51–152. MR 3468251
- G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 17, 8177–8179. MR 1180036, DOI 10.1073/pnas.89.17.8177
- Scott Edward Morrison, A diagrammatic category for the representation theory of $U_q(\mathfrak {sl}_n)$, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–University of California, Berkeley. MR 2710589
- Suho Oh, Positroids and Schubert matroids, J. Combin. Theory Ser. A 118 (2011), no. 8, 2426–2435. MR 2834184, DOI 10.1016/j.jcta.2011.06.006
- T. Kyle Petersen, Pavlo Pylyavskyy, and Brendon Rhoades, Promotion and cyclic sieving via webs, J. Algebraic Combin. 30 (2009), no. 1, 19–41. MR 2519848, DOI 10.1007/s10801-008-0150-3
- A. Postnikov, Total positivity, Grassmannians, and networks, preprint (2006) arXiv:0609764 [math.CO].
- Alexander Postnikov, David Speyer, and Lauren Williams, Matching polytopes, toric geometry, and the totally non-negative Grassmannian, J. Algebraic Combin. 30 (2009), no. 2, 173–191. MR 2525057, DOI 10.1007/s10801-008-0160-1
- K. Rietsch and L. Williams, Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, preprint (2017) arXiv:1712.00447 [math.AG].
- Julianna Tymoczko, A simple bijection between standard $3\times n$ tableaux and irreducible webs for $\mathfrak {sl}_3$, J. Algebraic Combin. 35 (2012), no. 4, 611–632. MR 2902703, DOI 10.1007/s10801-011-0317-1
- Bruce W. Westbury, Web bases for the general linear groups, J. Algebraic Combin. 35 (2012), no. 1, 93–107. MR 2873098, DOI 10.1007/s10801-011-0294-4
Additional Information
- Chris Fraser
- Affiliation: Department of Mathematical Sciences, Indiana Unversity–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202
- Email: chfraser@iupui.edu
- Thomas Lam
- Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
- MR Author ID: 679285
- ORCID: 0000-0003-2346-7685
- Email: tfylam@umich.edu
- Ian Le
- Affiliation: Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada, N2L 2Y5
- MR Author ID: 766385
- Email: ile@perimeterinstitute.ca
- Received by editor(s): June 23, 2017
- Published electronically: January 24, 2019
- Additional Notes: Some of the work took place at the Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.
The second author acknowledges support from the Simons Foundation under award number 341949 and from the NSF under agreement No. DMS-1464693. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6087-6124
- MSC (2010): Primary 05E10; Secondary 14M15, 20C30, 05C10
- DOI: https://doi.org/10.1090/tran/7641
- MathSciNet review: 3937319