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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 3, September 2019  pp. 291–303.

Polynomials associated with the fragments of coset diagrams

Authors:  Abdul Razaq, Qaiser Mushtaq and Awais Yousaf
Author institution:Department of Mathematics, University of Education Lahore, Jauharabad Campus, Pakistan

Summary:  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){q{1}}(xy{-1}){q{2}}, (xy{-1}){s{1}}(xy){s{2}}, where s{1}, s{2}, q{1}, q{2} ∈ 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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