Plucker

Plücker relations, separated set-systems, cubillages, quantum minors, and related topics:

V.I. Danilov, A.V. Karzanov, G.A. Koshevoy. On universal quadratic inequalities for minors of TNN matrices, arXiv:2506.03754 [math.CO], 10pp., 2025; https://doi.org/10.48550/arXiv.2506.03754 [PDF]; submitted
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Higher Bruhat orders of types B and C, J. of Algebraic Combin., 60, 2024, 241-264; DOI https://doi.org/10.1007/s10801-024-01334-x. (Draft in arXiv:2107.09462[math.CO], 2021, 22pp.; DOI:
https://doi.org/10.48550/arXiv.2107.09462. [PDF])
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, On interrelations between strongly, weakly and chord separated set-systems (a geometric approach), Journal of Algebraic Combinatorics, 54, 4, 2021, pp. 1299-1327. (Draft in arXiv:2506.03754 [math.CO], 10pp., 2025; https://doi.org/10.48550/arXiv.2506.03754. [PDF] )
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, The purity phenomenon for symmetric separated set-systems, Selecta Mathematics, New Series, 31, Article 31, 2025; DOI https://doi.org/10.1007/s00029-025-01025-w. (Draft in \textsl{Arxiv}:2007.02011v2[math.CO], 2020,22; DOI: https://doi.org/10.48550/arXiv.2007.02011.[PDF])
V.I. Danilov, A.V. Karzanov, Basic quadratic identities on quantum minors, Journal of Algebra, 578, 2021, pp. 303-318, DOI:10.1016/j.jalgebra.2021.02.027. [PDF]
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, The weak separation in higher dimensions, Journal of Combinatorial Theory, Series A, 180, 2021, Article 105392, https://doi.org/10.1016/j.jcta.2020.105392. [PDF]
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Cubillages of cyclic zonotopes, Uspekhi Matematicheskikh Nauk, 74, 6, 2019, pp. 55-118, in Russian; https://doi.org/10.4213/rm9879. (English Translation in Russian Math. Surveys, 74, 6, 2019, pp. 1013-1074. [PDF])
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Cubillages on cyclic zonotopes, membranes, and higher separation, arXiv:1810.05517[math.CO], 2018, 16pp. [PDF]
V.I. Danilov, A.V. Karzanov, On universal quadratic identities for minors of quantum matrices, Journal of Algebra, 488, 2017, pp. 145-200. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Combined tilings and separated set-systems, Selecta Mathematics, New Series, 23, 2017, pp. 1175–1203, DOI 10.1007/s00029-016-0264-8, 2016. [PDF]
V.I. Danilov and A.V. Karzanov, Two statements on path systems related to quantum minors, arXiv: 1611.00302[math.CO], 2016, 28 pp. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Ladder diagrams and weakly separated set-systems, J. of Algebra, Number Theory: Advances and Applications, 14, No. 2, 2015, 89-106. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Combined tilings and the purity phenomenon on separated set-systems, ArXiv:1401.6418 [math.CO], 2014, 36pp. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Tropical Plücker functions and Kashiwara crystals, In: Tropical and Idempotent Mathematics and Applications (International Workshop, Aug. 26–31, 2012, Independent Univ., Moscow, Russia) (eds. G.L. Litvinov and S.N. Sergeev), Contemporary Mathematics, 616, 2014, pp. 77–100. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Planar flows and quadratic relations over semirings, J. of Algebraic Combinatorics, 36, No.3, 2012, 441-474. (Draft in ArXiv:1102.2578v2[math.CO].) [PDF]. (See also [PDF])
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Systemy razdelennykh mnozhestv i ikh geometricheskie modeli, Uspekhi Matematicheskikh Nauk, 65, No.4, 2010, 132-217, in Russian [PDF]. (English transl.: Separated set-systems and their geometric models, in: Russian Math. Surveys, 65, No.4, 2010, 659-740.) (See also DOI 10.1016/j.dam.2011.08.001.)
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, On maximal weakly separated set-systems, J. of Algebraic Combinatorics, 32, 2010, 497-531. (Draft in ArXiv:0909.1423v2[math.CO].) [PDF] (See also [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Plücker environments, wiring and tiling diagrams, and weakly separated set-systems, Advances in Mathematics, 224, 2010, 1-44. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Tropical Plücker functions and their bases, in: Tropical and Idempotent Mathematics (ed. G.L. Litvinov and S.N. Sergeev), Contemporary Mathematics, 495, 2009, 127-158. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, On bases of tropical Plücker functions, ArXiv:0712.3996v2[math.CO], 2007, 44pp. [PDF]

Preprints, technical reports and extended abstracts:

V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Flips in symmetric separated set-systems, Arxiv:2102.08974[math.CO], 2021, 36 pp. [PDF]
V.I. Danilov, A.V. Karzanov, G.A. Koshevoy, Cubillages in odd dimensions, Seminaire Lotharingien de Combinatoire - FPSAC 2020 (Proc. of 32nd Conference on Formal Power Series and Algebraic Combinatorics), 84B, Article #10, 2020, 12 pp. [PDF]
V.I. Danilov and A.V. Karzanov, Path systems in certain planar graphs and quadratic identities for quantum minors, arXiv:1604.00338[math.QA], 2016, 33 pp. [PDF]
V.I. Danilov and A.V. Karzanov, On universal quadratic identities for minors of quantum matrices (extended abstract), Seminaire Lotharingien de Combinatoire - FPSAC 2017 (Proc. of the 29th Conference on Formal Power Series and Algebraic Combinatorics, July 9 -- 13, 2017, Queen Mary Univ. of London, UK), Issue 78B, 2017 Article #21, 12 pp. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, The purity of set-systems related to Grassmann necklaces, DMTCS Proceedings AT (FPSAC Symposium, Chicago, USA, 2014), 2014, pp. 193-204. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Tropical Plücker functions and Kashiwara crystals of types A, B, and C (extended abstract), In: Tropical and Idempotent Mathematics (International Workshop, Moscow, August 26-31, 2012), Inst. for Information Transmission, 2012, pp.66-77 [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Planar flows and quadratic relations over semirings, In: Tropical and Idempotent Mathematics (International Workshop, Moscow, August 26-31, 2012), Inst. for Information Transmission, 2012, pp.53-65 [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Coherent fans in the space of flows in directed graphs, DMTCS Proceedings AR (FPSAC Symposium, Nagoya Univ., Nagoya, Japan, 2012), 2012, pp. 483-494. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Generalized tilings and Plücker cluster algebras, Extended Abstract, In: Proc. of 7th International Conference on Lattice Path Combinatorics and Applications (Sienna, June 2010), 2010, 6pp. [PDF]
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, Plücker environments, wiring and tiling diagrams, and weakly separated set-systems, Preprint RIMS-1659, RIMS, Kyoto Univ., Kyoto, 2009, 42pp.
V.I. Danilov, A.V. Karzanov, and G.A. Koshevoy, On wiring and tiling diagrams related to bases of tropical Plücker functions, In: DMTCS Proceedings AK (FPSAC Symposium, Hagenberg, Austria, 2009), 2009, pp.327-338. [PDF]