Polynomials Associated with the Fragments of Coset Diagrams

Abstract

The coset diagrams for PS L (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 n of fragments such that each fragment .in n contains one vertex fixed by a pair of words (xy)q1 (xy-1 )q2 , (xy-1 y1 (xyY2 , where s1, s2, q1, q2 E z+, and prove Higman 's conjecture for the polynomials obtained from n. At the end, we answer the question; for a fixed degree n, how many polynomials have evolved from .