Tropical representations and identities of plactic monoids
HTML articles powered by AMS MathViewer
- by Marianne Johnson and Mark Kambites PDF
- Trans. Amer. Math. Soc. 374 (2021), 4423-4447 Request permission
Abstract:
We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank $n$ is satisfied by the monoid of $n \times n$ upper triangular tropical matrices. In particular this implies that the variety generated by the $3 \times 3$ upper triangular tropical matrices coincides with that generated by the plactic monoid of rank $3$, answering another question of Izhakian.References
- Marianne Akian, Stéphane Gaubert, and Alexander Guterman, Tropical polyhedra are equivalent to mean payoff games, Internat. J. Algebra Comput. 22 (2012), no. 1, 1250001, 43. MR 2900854, DOI 10.1142/S0218196711006674
- Xavier Allamigeon, Pascal Benchimol, Stéphane Gaubert, and Michael Joswig, Tropicalizing the simplex algorithm, SIAM J. Discrete Math. 29 (2015), no. 2, 751–795. MR 3336300, DOI 10.1137/130936464
- Alan J. Cain, Robert D. Gray, and António Malheiro, Finite Gröbner-Shirshov bases for plactic algebras and biautomatic structures for plactic monoids, J. Algebra 423 (2015), 37–53. MR 3283708, DOI 10.1016/j.jalgebra.2014.09.037
- Alan J. Cain, Robert D. Gray, and António Malheiro, Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$, J. Combin. Theory Ser. A 162 (2019), 406–466. MR 3876235, DOI 10.1016/j.jcta.2018.11.010
- Alan J. Cain and António Malheiro, Identities in plactic, hypoplactic, sylvester, Baxter, and related monoids, Electron. J. Combin. 25 (2018), no. 3, Paper No. 3.30, 19. MR 3853882, DOI 10.37236/6873
- A. J. Cain, G. Klein, L. Kubat, A. Malheiro and J. Okniński. A note on identities in plactic monoids and monoids of upper-triangular tropical matrices. Preprint at arXiv:1705.04596.
- Alan J. Cain, António Malheiro, and Fábio M. Silva, The monoids of the patience sorting algorithm, Internat. J. Algebra Comput. 29 (2019), no. 1, 85–125. MR 3918055, DOI 10.1142/S0218196718500649
- Ferran Cedó, Łukasz Kubat, and Jan Okniński, Irreducible representations of the plactic algebra of rank four, J. Algebra 488 (2017), 403–441. MR 3680925, DOI 10.1016/j.jalgebra.2016.03.046
- Filip Cools, Jan Draisma, Sam Payne, and Elina Robeva, A tropical proof of the Brill-Noether theorem, Adv. Math. 230 (2012), no. 2, 759–776. MR 2914965, DOI 10.1016/j.aim.2012.02.019
- Laure Daviaud, Marianne Johnson, and Mark Kambites, Identities in upper triangular tropical matrix semigroups and the bicyclic monoid, J. Algebra 501 (2018), 503–525. MR 3768147, DOI 10.1016/j.jalgebra.2017.12.032
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- Jeffrey Giansiracusa and Noah Giansiracusa, Equations of tropical varieties, Duke Math. J. 165 (2016), no. 18, 3379–3433. MR 3577368, DOI 10.1215/00127094-3645544
- S. Margolis and B. Steinberg, The algebra of the Catalan monoid as an incidence algebra: A simple proof, Preprint at arXiv:1806.06531.
- Pierre Guillon, Zur Izhakian, Jean Mairesse, and Glenn Merlet, The ultimate rank of tropical matrices, J. Algebra 437 (2015), 222–248. MR 3351964, DOI 10.1016/j.jalgebra.2015.02.026
- Christopher Hollings and Mark Kambites, Tropical matrix duality and Green’s $\scr D$ relation, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 520–538. MR 2980923, DOI 10.1112/jlms/jds015
- Zur Izhakian, Semigroup identities in the monoid of triangular tropical matrices, Semigroup Forum 88 (2014), no. 1, 145–161. MR 3164156, DOI 10.1007/s00233-013-9507-6
- Zur Izhakian and Stuart W. Margolis, Semigroup identities in the monoid of two-by-two tropical matrices, Semigroup Forum 80 (2010), no. 2, 191–218. MR 2601760, DOI 10.1007/s00233-009-9203-8
- Zur Izhakian, Semigroup identities of tropical matrix semigroups of maximal rank, Semigroup Forum 92 (2016), no. 3, 712–732. MR 3500264, DOI 10.1007/s00233-015-9765-6
- Zur Izhakian, Tropical plactic algebra, the cloaktic monoid, and semigroup representations, J. Algebra 524 (2019), 290–366. MR 3908267, DOI 10.1016/j.jalgebra.2018.12.014
- Zur Izhakian, Marianne Johnson, and Mark Kambites, Pure dimension and projectivity of tropical polytopes, Adv. Math. 303 (2016), 1236–1263. MR 3552549, DOI 10.1016/j.aim.2016.08.033
- Zur Izhakian, Marianne Johnson, and Mark Kambites, Tropical matrix groups, Semigroup Forum 96 (2018), no. 1, 178–196. MR 3750353, DOI 10.1007/s00233-017-9894-1
- Marianne Johnson and Peter Fenner, Identities in unitriangular and gossip monoids, Semigroup Forum 98 (2019), no. 2, 338–354. MR 3917968, DOI 10.1007/s00233-019-09996-x
- Marianne Johnson and Mark Kambites, Green’s $\scr {J}$-order and the rank of tropical matrices, J. Pure Appl. Algebra 217 (2013), no. 2, 280–292. MR 2969253, DOI 10.1016/j.jpaa.2012.06.003
- Marianne Johnson and Ngoc Mai Tran, Geometry and algorithms for upper triangular tropical matrix identities, J. Algebra 530 (2019), 470–507. MR 3957959, DOI 10.1016/j.jalgebra.2019.03.024
- M. Kambites, Free objects in triangular matrix varieties and quiver algebras over semirings, Preprint at arXiv:1904.06094.
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
- Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727. MR 272654
- Łukasz Kubat and Jan Okniński, Plactic algebra of rank 3, Semigroup Forum 84 (2012), no. 2, 241–266. MR 2898759, DOI 10.1007/s00233-011-9337-3
- Łukasz Kubat and Jan Okniński, Gröbner-Shirshov bases for plactic algebras, Algebra Colloq. 21 (2014), no. 4, 591–596. MR 3266510, DOI 10.1142/S1005386714000534
- Łukasz Kubat and Jan Okniński, Identities of the plactic monoid, Semigroup Forum 90 (2015), no. 1, 100–112. MR 3297811, DOI 10.1007/s00233-014-9609-9
- Alain Lascoux and Marcel-Paul Schützenberger, Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 7, A323–A324 (French, with English summary). MR 472993
- Alain Lascoux and Marcel-P. Schützenberger, Le monoïde plaxique, Noncommutative structures in algebra and geometric combinatorics (Naples, 1978) Quad. “Ricerca Sci.”, vol. 109, CNR, Rome, 1981, pp. 129–156 (French, with Italian summary). MR 646486
- M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski; With a preface by Berstel and Perrin. MR 1905123, DOI 10.1017/CBO9781107326019
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI 10.1090/S0894-0347-05-00477-7
- Jan Okniński, Identities of the semigroup of upper triangular tropical matrices, Comm. Algebra 43 (2015), no. 10, 4422–4426. MR 3366585, DOI 10.1080/00927872.2014.946141
- J. Okniński, Plactic monoids satisfy nontrivial identities, Preprint at arXiv:1902.06251v2. Retraction notice at arXiv:1902.06251v3.
- Jean-Eric Pin, Tropical semirings, Idempotency (Bristol, 1994) Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998, pp. 50–69. MR 1608374, DOI 10.1017/CBO9780511662508.004
- C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 (1961), 179–191. MR 121305, DOI 10.4153/CJM-1961-015-3
- M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 59–113 (French). MR 0498826
- Yaroslav Shitov, The complexity of tropical matrix factorization, Adv. Math. 254 (2014), 138–156. MR 3161095, DOI 10.1016/j.aim.2013.12.013
- Imre Simon, On semigroups of matrices over the tropical semiring, RAIRO Inform. Théor. Appl. 28 (1994), no. 3-4, 277–294. MR 1282449, DOI 10.1051/ita/1994283-402771
- M. Taylor, On upper triangular tropical matrix semigroups, tropical matrix identities and T-modules, Ph.D. thesis, University of Manchester, 2017.
Additional Information
- Marianne Johnson
- Affiliation: Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
- MR Author ID: 800904
- ORCID: 0000-0003-4059-845X
- Email: Marianne.Johnson@manchester.ac.uk
- Mark Kambites
- Affiliation: Department of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
- MR Author ID: 760844
- Email: Mark.Kambites@manchester.ac.uk
- Received by editor(s): May 27, 2020
- Received by editor(s) in revised form: November 9, 2020
- Published electronically: March 30, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4423-4447
- MSC (2020): Primary 20M07, 20M30, 05E99, 12K10, 16Y60
- DOI: https://doi.org/10.1090/tran/8355
- MathSciNet review: 4251234