Journal of the Ramanujan Mathematical Society

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Polynomials Associated with the Fragments of Coset Diagrams


Affiliations
1 Department of Mathematics, University of Education Lahore, Jauharabad Campus, Pakistan
2 The Islamia University of Bahawalpur, Pakistan
3 Department of Mathematics, The Islamia University of Bahawalpur, Pakistan
     

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The coset diagrams for PSL(2, ℤ) are composed of fragments, and the fragments are further composed of circuits. Mushtaq has found that, the condition for the existence of a fragment in coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree, there are finite number of such polynomials. In this paper, we consider a family Ω of fragments such that each fragment in Ω contains one vertex fixed by a pair of words (xy)q1 (xy−1)q2, (xy−1)s1 (xy)s2 , where s1, s2, q1, q2 ∈ ℤ+, and prove Higman’s conjecture for the polynomials obtained from Ω. At the end, we answer the question; for a fixed degree n, how many polynomials are evolved from Ω.
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  • Polynomials Associated with the Fragments of Coset Diagrams

Abstract Views: 233  |  PDF Views: 0

Authors

Abdul Razaq
Department of Mathematics, University of Education Lahore, Jauharabad Campus, Pakistan
Qaiser Mushtaq
The Islamia University of Bahawalpur, Pakistan
Awais Yousaf
Department of Mathematics, The Islamia University of Bahawalpur, Pakistan

Abstract


The coset diagrams for PSL(2, ℤ) are composed of fragments, and the fragments are further composed of circuits. Mushtaq has found that, the condition for the existence of a fragment in coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree, there are finite number of such polynomials. In this paper, we consider a family Ω of fragments such that each fragment in Ω contains one vertex fixed by a pair of words (xy)q1 (xy−1)q2, (xy−1)s1 (xy)s2 , where s1, s2, q1, q2 ∈ ℤ+, and prove Higman’s conjecture for the polynomials obtained from Ω. At the end, we answer the question; for a fixed degree n, how many polynomials are evolved from Ω.

References