Rocky Mountain Journal of Mathematics

Gröbner-Shirshov bases of some Weyl groups

Eylem Güzel Karpuz, Firat Ateş, and A. Sinan Çevik

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Abstract

In this paper, we obtain Gr\"{o}bner-Shirshov (non-commutative) bases for the $n$-extended affine Weyl group $\widetilde{W}$ of type $A_1$, elliptic Weyl groups of types $A_{1}^{(1,1)}$, $A_{1}^{(1,1)^{*}}$ and the $2$-extended affine Weyl group of type $A_{2}^{(1,1)}$ with a generator system of a $2$-toroidal sense. It gives a new algorithm for getting normal forms of elements of these groups and hence a new algorithm for solving the word problem in these groups.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 4 (2015), 1165-1175.

Dates
First available in Project Euclid: 2 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1446472428

Digital Object Identifier
doi:10.1216/RMJ-2015-45-4-1165

Mathematical Reviews number (MathSciNet)
MR3418188

Zentralblatt MATH identifier
0895.16020

Subjects
Primary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

Keywords
Gröbner-Shirshov basis presentation Weyl group

Citation

Karpuz, Eylem Güzel; Ateş, Firat; Çevik, A. Sinan. Gröbner-Shirshov bases of some Weyl groups. Rocky Mountain J. Math. 45 (2015), no. 4, 1165--1175. doi:10.1216/RMJ-2015-45-4-1165. https://projecteuclid.org/euclid.rmjm/1446472428


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References

  • S.I. Adian and V.G. Durnev, Decision problems for groups and semigroups, Russian Math. Surv. 55 (2000), 207–296.
  • F. Ateş, E.G. Karpuz, C. Kocapinar and A.S. Çevik, Gröbner-Shirshov bases of some monoids, Discr. Math. 311 (2011), 1064–1071.
  • G.M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218.
  • L.A. Bokut, Imbedding into simple associative algebras, Alg. Logic 15 (1976), 117–142.
  • L.A. Bokut, Gröbner-Shirshov basis for the Braid group in the Artin-Garside generators, J. Symb. Comp. 43 (2008), 397–405.
  • ––––, Gröbner-Shirshov basis for the Braid group in the Birman–Ko–Lee generators, J. Alg. 321 (2009), 361–376.
  • L.A. Bokut and A. Vesnin, Gröbner-Shirshov bases for some Braid groups, J. Symb. Comp. 41 (2006), 357–371.
  • B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal, Ph.D. thesis, University of Innsbruck, Innsbruck, Austria, 1965.
  • Y. Chen and C. Zhong, Gröbner-Shirshov bases for HNN extentions of groups and for the alternating group, Comm. Alg. 36 (2008), 94–103.
  • E.G. Karpuz, Gröbner-Shirshov bases of some semigroup constructions, Alg. Colloq. 22 (2015), 35–46.
  • C. Kocapinar, E.G. Karpuz, F. Ateş and A.S. Çevik, Gröbner-Shirshov bases of the generalized Bruck-Reilly $*$-extension, Alg. Colloq. 19 (2012), 813–820.
  • K. Saito, Extended affine root systems I, Coxeter transformations, Publ. Res. Inst. Math. Sci. 21 (1985), 75–179.
  • K. Saito and T. Takebayashi, Extended affine root systmes III, Elliptic Weyl groups, Publ. Res. Inst. Math. Sci. 33 (1997), 301–329.
  • A.I. Shirshov, Some algorithmic problems for Lie algebras, Siber. Math. J. 3 (1962), 292–296.
  • T. Takebayashi, Defining relations of the Weyl groups for extended affine root systems $A_{l}^{(1,1)}$, $B_{l}^{(1,1)}$, $C_{l}^{(1,1)}$, $D_{l}^{(1,1)}$, J. Alg. 168 (1994), 810–827.
  • ––––, Relations of the Weyl groups of extended affine root systems $A_{l}^{(1,1)}$, $B_{l}^{(1,1)}$, $C_{l}^{(1,1)}$, $D_{l}^{(1,1)}$, Proc. Japan Acad. 71 (1995), 123–124.
  • ––––, The growth series of the $n$-extended affine Weyl group of type $A_1$, Proc. Japan Acad. 81 (2005), 51–56.
  • ––––, Poincare series of the Weyl group of the elliptic root systems $A_{l}^{(1,1)}$, $A_{l}^{(1,1)*}$ and $A_{2}^{(1,1)}$, J. Alg. Comb. 17 (2003), 211–223.