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Yousaf, Awais
- Polynomials Associated with the Fragments of Coset Diagrams
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1 Department of Mathematics, University of Education Lahore, Jauharabad Campus, PK
2 The Islamia University of Bahawalpur, PK
3 Department of Mathematics, The Islamia University of Bahawalpur, PK
1 Department of Mathematics, University of Education Lahore, Jauharabad Campus, PK
2 The Islamia University of Bahawalpur, PK
3 Department of Mathematics, The Islamia University of Bahawalpur, PK
Source
Journal of the Ramanujan Mathematical Society, Vol 34, No 3 (2019), Pagination: 291-303Abstract
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
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