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Year 2019, Volume: 7 Issue: 1, 79 - 90, 15.04.2019

Abstract

References

  • [1] Adian, S. I., Durnev, V. G., Decision problems for groups and semigroups, Russian Math. Surveys Vol:55, No.2 (2000), 207-296.
  • [2] Ateş, F., Çevik, A. S. , Karpuz, E. G., Gr¨obner-Shirshov basis for the singular part of the Brauer semigroup, Turkish Journal of Math. Vol:42 (2018), 1338-1347.
  • [3] Ateş, F., Karpuz, E. G., Kocapinar, C. Çevik, A. S., Gr¨obner-Shirshov bases of some monoids, Discrete Math. Vol:311 (2011), 1064-1071.
  • [4] Bergman, G. M., The diamond lemma for ring theory, Adv. Math. Vol:29 (1978), 178-218.
  • [5] Bessis, D., Michel, J., Explicit presentations for exceptional Braid groups, Experimental Math. Vol:13, No.3 (2004), 257-266.
  • [6] Bokut, L. A., Imbedding into simple associative algebras, Algebra and Logic Vol:15 (1976), 117-142.
  • [7] Bokut, L. A., Vesnin, A., Gr¨obner-Shirshov bases for some Braid groups, Journal of Symbolic Comput. Vol:41 (2006), 357-371.
  • [8] Bokut, L. A., Gr¨obner-Shirshov basis for the Braid group in the Birman-Ko-Lee generators, Journal Algebra Vol:321 (2009), 361-376.
  • [9] Bokut, L. A., Gr¨obner-Shirshov basis for the Braid group in the Artin-Garside generators, Journal of Symbolic Comput. Vol:43 (2008), 397-405.
  • [10] Buchberger, B., An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Ideal, Ph.D. Thesis, University of Innsbruck, 1965.
  • [11] Chen, Y., Zhong, C., Gröbner-Shirshov bases for HNN extentions of groups and for the Alternating group, Comm. in Algebra Vol:36 (2008), 94-103.
  • [12] Cohen, A.M., Finite complex reflection groups, Ann. Scient. E0 c. Norm. Sup. 4e Se0 rie. Vol:9 (1976), 379-436.
  • [13] Howlett, R.B., Shi, J.Y., On regularity of finite reflection groups, Manuscripta Mathematica Vol:102, No.3 (2000), 325-333.
  • [14] Karpuz, E. G., Gr¨obner-Shirshov bases of some semigroup constructions, Algebra Colloquium Vol:22, No.1 (2015), 35-46.
  • [15] Karpuz, E. G., Ates¸, F., C¸ evik, A. S., Gr¨obner-Shirshov bases of some weyl groups, Rocky Mountain Journal of Math. Vol:45, No.4 (2015), 1165-1175.
  • [16] Karpuz, E. G., C¸ evik, A. S., Ates¸, F., Koppitz, J., Gr¨obner-Shirshov bases and embedding of a semigroup in a group, Adv. Studies Contemp. Math. vol:25, No.4 (2015), 537-545.
  • [17] Karpuz, E. G., Ates¸, F., Urlu, N., C¸ evik, A. S., Cang¨ul, I.N., A Note on the Gr¨obner-Shirshov bases over ad-hoc extensions of groups, Filomat Vol:30, No.4 (2016), 1037-1043.
  • [18] Kocapinar, C., Karpuz, E.G., Ates¸, F., C¸ evik, A.S., Gr¨obner-Shirshov bases of the generalized Bruck-Reilly -extension, Algebra Colloquium Vol:19(Spec 1) (2012), 813-820.
  • [19] Shephard, G. C., Todd, J. A., Finite unitary reflection groups, Canad. J. Math. Vol:6 (1954), 274-304.
  • [20] Shi, J.Y., Certain imprimitive reflection groups and their generic versions, Trans. Am. Math. Soc. Vol:354 (2002), 2115-2129.
  • [21] Shi, J.Y., Simple root systems and presentations for certain complex reflection groups, Comm. in Algebra Vol:33 (2005), 1765-1783.
  • [22] Shirshov, A. I., Some algorithmic problems for Lie algebras, Siberian Math. J. Vol:3 (1962), 292-296.

Gröbner-Shirshov Basis for Complex Reflection Group

Year 2019, Volume: 7 Issue: 1, 79 - 90, 15.04.2019

Abstract

The aim of this paper is to obtain a (non-commutative) Gröbner-Shirshov basis for the braid group associated with the complex reflection group $G_{24}$. This gives us an opportunity to get normal forms of the elements of group $G_{24}$, which represent a new and effective algorithm to solve the word problem over it.



References

  • [1] Adian, S. I., Durnev, V. G., Decision problems for groups and semigroups, Russian Math. Surveys Vol:55, No.2 (2000), 207-296.
  • [2] Ateş, F., Çevik, A. S. , Karpuz, E. G., Gr¨obner-Shirshov basis for the singular part of the Brauer semigroup, Turkish Journal of Math. Vol:42 (2018), 1338-1347.
  • [3] Ateş, F., Karpuz, E. G., Kocapinar, C. Çevik, A. S., Gr¨obner-Shirshov bases of some monoids, Discrete Math. Vol:311 (2011), 1064-1071.
  • [4] Bergman, G. M., The diamond lemma for ring theory, Adv. Math. Vol:29 (1978), 178-218.
  • [5] Bessis, D., Michel, J., Explicit presentations for exceptional Braid groups, Experimental Math. Vol:13, No.3 (2004), 257-266.
  • [6] Bokut, L. A., Imbedding into simple associative algebras, Algebra and Logic Vol:15 (1976), 117-142.
  • [7] Bokut, L. A., Vesnin, A., Gr¨obner-Shirshov bases for some Braid groups, Journal of Symbolic Comput. Vol:41 (2006), 357-371.
  • [8] Bokut, L. A., Gr¨obner-Shirshov basis for the Braid group in the Birman-Ko-Lee generators, Journal Algebra Vol:321 (2009), 361-376.
  • [9] Bokut, L. A., Gr¨obner-Shirshov basis for the Braid group in the Artin-Garside generators, Journal of Symbolic Comput. Vol:43 (2008), 397-405.
  • [10] Buchberger, B., An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Ideal, Ph.D. Thesis, University of Innsbruck, 1965.
  • [11] Chen, Y., Zhong, C., Gröbner-Shirshov bases for HNN extentions of groups and for the Alternating group, Comm. in Algebra Vol:36 (2008), 94-103.
  • [12] Cohen, A.M., Finite complex reflection groups, Ann. Scient. E0 c. Norm. Sup. 4e Se0 rie. Vol:9 (1976), 379-436.
  • [13] Howlett, R.B., Shi, J.Y., On regularity of finite reflection groups, Manuscripta Mathematica Vol:102, No.3 (2000), 325-333.
  • [14] Karpuz, E. G., Gr¨obner-Shirshov bases of some semigroup constructions, Algebra Colloquium Vol:22, No.1 (2015), 35-46.
  • [15] Karpuz, E. G., Ates¸, F., C¸ evik, A. S., Gr¨obner-Shirshov bases of some weyl groups, Rocky Mountain Journal of Math. Vol:45, No.4 (2015), 1165-1175.
  • [16] Karpuz, E. G., C¸ evik, A. S., Ates¸, F., Koppitz, J., Gr¨obner-Shirshov bases and embedding of a semigroup in a group, Adv. Studies Contemp. Math. vol:25, No.4 (2015), 537-545.
  • [17] Karpuz, E. G., Ates¸, F., Urlu, N., C¸ evik, A. S., Cang¨ul, I.N., A Note on the Gr¨obner-Shirshov bases over ad-hoc extensions of groups, Filomat Vol:30, No.4 (2016), 1037-1043.
  • [18] Kocapinar, C., Karpuz, E.G., Ates¸, F., C¸ evik, A.S., Gr¨obner-Shirshov bases of the generalized Bruck-Reilly -extension, Algebra Colloquium Vol:19(Spec 1) (2012), 813-820.
  • [19] Shephard, G. C., Todd, J. A., Finite unitary reflection groups, Canad. J. Math. Vol:6 (1954), 274-304.
  • [20] Shi, J.Y., Certain imprimitive reflection groups and their generic versions, Trans. Am. Math. Soc. Vol:354 (2002), 2115-2129.
  • [21] Shi, J.Y., Simple root systems and presentations for certain complex reflection groups, Comm. in Algebra Vol:33 (2005), 1765-1783.
  • [22] Shirshov, A. I., Some algorithmic problems for Lie algebras, Siberian Math. J. Vol:3 (1962), 292-296.
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Eylem Güzel Karpuz 0000-0002-7111-3462

Nurten Urlu Özalan 0000-0002-3022-350X

Ahmet Sinan Çevik 0000-0002-7539-5065

Publication Date April 15, 2019
Submission Date November 26, 2018
Acceptance Date March 27, 2019
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Güzel Karpuz, E., Urlu Özalan, N., & Çevik, A. S. (2019). Gröbner-Shirshov Basis for Complex Reflection Group. Konuralp Journal of Mathematics, 7(1), 79-90.
AMA Güzel Karpuz E, Urlu Özalan N, Çevik AS. Gröbner-Shirshov Basis for Complex Reflection Group. Konuralp J. Math. April 2019;7(1):79-90.
Chicago Güzel Karpuz, Eylem, Nurten Urlu Özalan, and Ahmet Sinan Çevik. “Gröbner-Shirshov Basis for Complex Reflection Group”. Konuralp Journal of Mathematics 7, no. 1 (April 2019): 79-90.
EndNote Güzel Karpuz E, Urlu Özalan N, Çevik AS (April 1, 2019) Gröbner-Shirshov Basis for Complex Reflection Group. Konuralp Journal of Mathematics 7 1 79–90.
IEEE E. Güzel Karpuz, N. Urlu Özalan, and A. S. Çevik, “Gröbner-Shirshov Basis for Complex Reflection Group”, Konuralp J. Math., vol. 7, no. 1, pp. 79–90, 2019.
ISNAD Güzel Karpuz, Eylem et al. “Gröbner-Shirshov Basis for Complex Reflection Group”. Konuralp Journal of Mathematics 7/1 (April 2019), 79-90.
JAMA Güzel Karpuz E, Urlu Özalan N, Çevik AS. Gröbner-Shirshov Basis for Complex Reflection Group. Konuralp J. Math. 2019;7:79–90.
MLA Güzel Karpuz, Eylem et al. “Gröbner-Shirshov Basis for Complex Reflection Group”. Konuralp Journal of Mathematics, vol. 7, no. 1, 2019, pp. 79-90.
Vancouver Güzel Karpuz E, Urlu Özalan N, Çevik AS. Gröbner-Shirshov Basis for Complex Reflection Group. Konuralp J. Math. 2019;7(1):79-90.
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