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GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$

Year 2024, Volume: 7 Issue: To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI", 86 - 101, 29.12.2024

Uğur Ustaoğlu

https://doi.org/10.33773/jum.1562793

Abstract

In this paper, we will investigate the infinite Coxeter groups $\widetilde{B}_n$ and $\widetilde{D}_n$. Their Gr\"{o}bner-Shirshov bases and classifications of normal forms are achieved by leveraging results from the infinite Coxeter groups of types $\widetilde{C}_n$. Additionally, a new algorithm is presented for obtaining normal forms of elements within these groups.

Keywords

Gr\"{o}bner-Shirshov Bases coxeter groups normal forms

References

  • L. A. Bokut, L. S. Shiao, Grobner bases for Coxeter groups, Comm. in Algebra, Vol.29, pp. 4305-4319, (2001).
  • A. Bjorner, F. Brenti, Combinatoric of Coxeter Groups, Springer-Verlag Berlin Heidelberg, New York, (2005).
  • O. Svechkarenko, Grobner-Shirshov Bases for the Coxeter Group E8, Master Thesis. Novosibirsk State University, (2007).
  • D. V. Lee, Grobner-Shirshov Bases and Normal Forms for the Coxeter Groups E6 and E7, Advances in Algebra and Combinatorics, World Scientific, Hackensack, pp.243-255, (2008). DOI number 10.1081/AGB-100106002
  • E. G. Karpuz and A. S. Cevik, Grobner-Shirshov bases for Extendes Modular, Extended Hecke, and Picard Groups, Mathematical Notes, Vol.92, No.5, pp.636-642, (2012). Published in Russian in Matematicheskie Zametki, Vol.92, No.5, pp.699-706, (2012). DOI number 10.1134/S0001434612110065
  • E. G. Karpuz, F. Ates, A. S. Cevik, I. N. Cangul, The Graph Based on Grobner-Shirshov Bases of Groups, Fixed Point Theory and Applications, Vol.71, (2013).DOI number 10.1186/1687-1812-2013-71
  • U. Ustaoglu, Grobner-Shirshov Bases Reduced Forms for Affine Weyl Groups, Master Thesis. Bolu Abant Izzet Baysal University, (2012).
  • E. Yilmaz, C. Ozel, U. Ustaoglu, Grobner-Shirshov Basis and Reduced Words for the Affine Weyl Group An, Journal of Algebra and Its Applications, Vol.13, No.6, (2014). DOI number 10.1142/S0219498814500054
  • E. G. Karpuz, F. Ates, A. S. Cevik, Grobner-Shirshov Bases of some Weyl groups, Rocky Mountain Journal of Mathematics, Vol.45, No.4, pp.1165-1175, (2015). DOI number 10.1216/RMJ-2015-45-4-1165
  • D. Lee, Standard monomials for the Weyl group F4, Journal of Algebra and Its Application, Vol.15, No.8, (2016).DOI number 10.1142/S0219498816501462
  • F. Ates, A. S. Cevik and E. G. Karpuz, Grobner-Shirshov Basis for the Singular Part of the Brauer Semigroup, Turkish Journal of Mathematics, Vol.42, pp.1338-1347, (2018).

Year 2024, Volume: 7 Issue: To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI", 86 - 101, 29.12.2024

Uğur Ustaoğlu

https://doi.org/10.33773/jum.1562793

Abstract

References

  • L. A. Bokut, L. S. Shiao, Grobner bases for Coxeter groups, Comm. in Algebra, Vol.29, pp. 4305-4319, (2001).
  • A. Bjorner, F. Brenti, Combinatoric of Coxeter Groups, Springer-Verlag Berlin Heidelberg, New York, (2005).
  • O. Svechkarenko, Grobner-Shirshov Bases for the Coxeter Group E8, Master Thesis. Novosibirsk State University, (2007).
  • D. V. Lee, Grobner-Shirshov Bases and Normal Forms for the Coxeter Groups E6 and E7, Advances in Algebra and Combinatorics, World Scientific, Hackensack, pp.243-255, (2008). DOI number 10.1081/AGB-100106002
  • E. G. Karpuz and A. S. Cevik, Grobner-Shirshov bases for Extendes Modular, Extended Hecke, and Picard Groups, Mathematical Notes, Vol.92, No.5, pp.636-642, (2012). Published in Russian in Matematicheskie Zametki, Vol.92, No.5, pp.699-706, (2012). DOI number 10.1134/S0001434612110065
  • E. G. Karpuz, F. Ates, A. S. Cevik, I. N. Cangul, The Graph Based on Grobner-Shirshov Bases of Groups, Fixed Point Theory and Applications, Vol.71, (2013).DOI number 10.1186/1687-1812-2013-71
  • U. Ustaoglu, Grobner-Shirshov Bases Reduced Forms for Affine Weyl Groups, Master Thesis. Bolu Abant Izzet Baysal University, (2012).
  • E. Yilmaz, C. Ozel, U. Ustaoglu, Grobner-Shirshov Basis and Reduced Words for the Affine Weyl Group An, Journal of Algebra and Its Applications, Vol.13, No.6, (2014). DOI number 10.1142/S0219498814500054
  • E. G. Karpuz, F. Ates, A. S. Cevik, Grobner-Shirshov Bases of some Weyl groups, Rocky Mountain Journal of Mathematics, Vol.45, No.4, pp.1165-1175, (2015). DOI number 10.1216/RMJ-2015-45-4-1165
  • D. Lee, Standard monomials for the Weyl group F4, Journal of Algebra and Its Application, Vol.15, No.8, (2016).DOI number 10.1142/S0219498816501462
  • F. Ates, A. S. Cevik and E. G. Karpuz, Grobner-Shirshov Basis for the Singular Part of the Brauer Semigroup, Turkish Journal of Mathematics, Vol.42, pp.1338-1347, (2018).
There are 11 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory, Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)
Journal Section Research Article
Authors

Uğur Ustaoğlu 0000-0001-7375-0652

Publication Date December 29, 2024
Submission Date October 7, 2024
Acceptance Date November 20, 2024
Published in Issue Year 2024 Volume: 7 Issue: To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI"

Cite

APA Ustaoğlu, U. (2024). GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$. Journal of Universal Mathematics, 7(To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI"), 86-101. https://doi.org/10.33773/jum.1562793
AMA Ustaoğlu U. GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$. JUM. December 2024;7(To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI"):86-101. doi:10.33773/jum.1562793
Chicago Ustaoğlu, Uğur. “GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$”. Journal of Universal Mathematics 7, no. To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI" (December 2024): 86-101. https://doi.org/10.33773/jum.1562793.
EndNote Ustaoğlu U (December 1, 2024) GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$. Journal of Universal Mathematics 7 To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI" 86–101.
IEEE U. Ustaoğlu, “GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$”, JUM, vol. 7, no. To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI", pp. 86–101, 2024, doi: 10.33773/jum.1562793.
ISNAD Ustaoğlu, Uğur. “GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$”. Journal of Universal Mathematics 7/To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI" (December 2024), 86-101. https://doi.org/10.33773/jum.1562793.
JAMA Ustaoğlu U. GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$. JUM. 2024;7:86–101.
MLA Ustaoğlu, Uğur. “GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$”. Journal of Universal Mathematics, vol. 7, no. To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI", 2024, pp. 86-101, doi:10.33773/jum.1562793.
Vancouver Ustaoğlu U. GRÖBNER-SHIRSHOV BASES AND NORMAL FORMS FOR THE INFINITE COXETER GROUPS OF TYPES $\widetilde{B}_{n}$ and $\widetilde{D}_n$. JUM. 2024;7(To memory "Assoc. Prof. Dr. Zeynep AKDEMİRCİ ŞANLI"):86-101.
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