Startseite The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
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The Gelfand–Kirillov dimension of Hecke–Kiselman algebras

  • Magdalena Wiertel ORCID logo EMAIL logo
Veröffentlicht/Copyright: 27. Januar 2023

Abstract

Hecke–Kiselman algebras A Θ , over a field 𝕂 , associated to finite oriented graphs Θ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand–Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph Θ that determines the dimension of A Θ is found. Namely, we prove that the Gelfand–Kirillov dimension of A Θ is the sum of the number of cyclic subgraphs of Θ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.


Communicated by Manfred Droste


Funding statement: This work is supported by grant 2021/41/N/ST1/03082 of the National Science Centre (Poland). For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

Acknowledgements

I am very grateful to Jan Okniński for careful reading and many suggestions on the earlier versions of the paper, as well as his continuous support. I would also like to thank Arkadiusz Męcel for fruitful discussions about the research topic.

References

[1] R. Aragona and A. D’Andrea, Hecke–Kiselman monoids of small cardinality, Semigroup Forum 86 (2013), no. 1, 32–40. 10.1007/s00233-012-9422-2Suche in Google Scholar

[2] D. N. Ashikhmin, M. V. Volkov and W. T. Zhang, The finite basis problem for Kiselman monoids, Demonstr. Math. 48 (2015), no. 4, 475–492. 10.1515/dema-2015-0035Suche in Google Scholar

[3] G. M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), no. 2, 178–218. 10.1016/0001-8708(78)90010-5Suche in Google Scholar

[4] T. Denton, F. Hivert, A. Schilling and N. M. Thiéry, On the representation theory of finite 𝒥 -trivial monoids, Sém. Lothar. Combin. 64 (2011), Article ID B64d. Suche in Google Scholar

[5] L. Forsberg, Effective representations of Hecke–Kiselman monoids of type A, preprint (2014), https://arxiv.org/abs/1205.0676. Suche in Google Scholar

[6] O. Ganyushkin and V. Mazorchuk, On Kiselman quotients of 0-Hecke monoids, Int. Electron. J. Algebra 10 (2011), 174–191. Suche in Google Scholar

[7] J. Huang, A tableau approach to the representation theory of 0-Hecke algebras, Ann. Comb. 20 (2016), no. 4, 831–868. 10.1007/s00026-016-0338-5Suche in Google Scholar

[8] G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand–Kirillov dimension, Res. Notes Math. 116, Pitman, Boston, 1985. Suche in Google Scholar

[9] G. Kudryavtseva and V. Mazorchuk, On Kiselman’s semigroup, Yokohama Math. J. 55 (2009), no. 1, 21–46. Suche in Google Scholar

[10] E. Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, J. Combin. Theory Ser. A 161 (2019), 178–219. 10.1016/j.jcta.2018.07.013Suche in Google Scholar

[11] A. Mȩcel and J. Okniński, Gröbner basis and the automaton property of Hecke–Kiselman algebras, Semigroup Forum 99 (2019), no. 2, 447–464. 10.1007/s00233-019-10029-wSuche in Google Scholar

[12] A. Mȩcel and J. Okniński, Growth alternative for Hecke–Kiselman monoids, Publ. Mat. 63 (2019), no. 1, 219–240. 10.5565/PUBLMAT6311907Suche in Google Scholar

[13] J. Okniński and M. Wiertel, Combinatorics and structure of Hecke–Kiselman algebras, Commun. Contemp. Math. 22 (2020), no. 7, Article ID 2050022. 10.1142/S0219199720500224Suche in Google Scholar

[14] J. C. Rosales and P. A. García-Sánchez, Numerical Semigroups, Dev. Math. 20, Springer, New York, 2009. 10.1007/978-1-4419-0160-6Suche in Google Scholar

[15] A. Szilard, S. Yu, K. Zhang and J. Shallit, Characterizing regular languages with polynomial densities, Mathematical Foundations of Computer Science 1992), Lecture Notes in Comput. Sci. 629, Springer, Berlin (1992), 494–503. 10.1007/3-540-55808-X_48Suche in Google Scholar

[16] V. A. Ufnarovskij, Combinatorial and asymptotic methods in algebra, Algebra, VI, Encyclopaedia Math. Sci. 57, Springer, Berlin (1995), 1–196. 10.1007/978-3-662-06292-0_1Suche in Google Scholar

Received: 2022-07-08
Revised: 2022-12-21
Published Online: 2023-01-27
Published in Print: 2023-03-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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