Abstract
The structure of the algebra K[M] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K[M] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M. Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K[M] are described. All these representations are monomial. As an application, a new proof of the semiprimitivity of K[M] is given.
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References
Bergman, G.M.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)
Cassaigne, J., Espie, M., Krob, D., Novelli, J.-C., Hivert, F.: The Chinese monoid. Int. J. Algebra Comput. 11, 301–334 (2001)
Cedó, F., Okniński, J.: Plactic algebras. J. Algebra 274, 97–117 (2004)
Cedó, F., Okniński, J.: Minimal spectrum and the radical of Chinese algebras. Preprint (2010)
Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. 1. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1977)
Duchamp, G., Krob, D.: Plactic-growth-like monoids. In: Words, Languages and Combinatorics II, pp. 124–142. World Scientific, Singapore (1994)
Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, New York (1997)
Gateva-Ivanova, T.: A combinatorial approach to set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45, 3828–3858 (2004)
Jaszuńska, J., Okniński, J.: Chinese algebras of rank 3. Commun. Algebra 34, 2745–2754 (2006)
Jaszuńska, J., Okniński, J.: Structure of Chinese algebras. J. Algebra (to appear), available at http://arxiv.org/abs/1009.5847v1
Jespers, E., Okniński, J.: Noetherian Semigroup Algebras. Algebra and Applications, vol. 7. Springer, Dordrecht (2007)
Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. Graduate Studies in Mathematics, vol. 22. American Mathematical Society, Providence (2000)
Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131. Springer, New York (2001)
Lascoux, A., Leclerc, B., Thibon, J.-Y.: The plactic monoid. In: Algebraic Combinatorics on Words, pp. 164–196. Cambridge University Press, Cambridge (2002)
Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. In: Noncommutative Structures in Algebra and Geometric Combinatorics, pp. 129–156. Dekker, Naples (1978)
Okniński, J.: Semigroup Algebras. Pure and Applied Mathematics, vol. 138. Dekker, New York (1991)
Passman, D.S.: The Algebraic Structure of Group Rings. Wiley-Interscience, New York (1977)
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Communicated by Mohan S. Putcha.
The second author is supported by MNiSW research grant N201 420539 (Poland).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kubat, Ł., Okniński, J. Plactic algebra of rank 3. Semigroup Forum 84, 241–266 (2012). https://doi.org/10.1007/s00233-011-9337-3
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DOI: https://doi.org/10.1007/s00233-011-9337-3