Congruence classes of presentations for the complex reflection groups \(G(m,p,n)\).
Let \(V\) be a complex vector space of dimension \(n\). A reflection on \(V\) is a linear transformation on \(V\) of finite order with exactly \(n-1\) eigenvalues equal to 1. A reflection group \(G\) on \(V\) is a finite group generated by reflections on \(V\). A reflection group \(G\) in \(V\) is imprimitive, if \(G\) acts on \(V\) irreducibly and there exists a decomposition \(V=V_1\oplus\cdots\oplus V_r\) of non-trivial proper subspaces \(V_i\), \(1\leq i\leq r\), of \(V\) such that \(G\) permutes the set \(\{V_i\mid 1\leq i\leq r\}\). For a reflection group \(G\), a presentation of \(G\) by generators and relations is by definition a pair \((S,P)\), where (1) \(S\) is a finite generator set for \(G\) which consists of reflections, and \(S\) has minimal cardinality with this property. (2) \(P\) is a finite set of relations on \(S\), and any other relation on \(S\) is a consequence of the relations in \(P\). A presentation \((S,P)\) of \(G\) is essential if \((S,P_0)\) is not a presentation of \(G\) for any proper subset \(P_0\) of \(P\). Two presentations \((S,P)\) and \((S',P')\) for \(G\) are congruent, if there exists a bijection \(\eta\colon S\to S'\) such that for any \(s,t\in S\), \(\langle s,t\rangle\cong\langle\eta(s),\eta(t)\rangle\), where the notation \(\langle x,y\rangle\) stands for the group generated by \(x,y\). In this paper, the author gives an explicit description in terms of rooted graphs for representatives of all the congruence classes of presentations for the imprimitive complex reflection group \(G(m,p,n)\). Also the author shows that \((S,P_S)\) forms a presentation of \(G(m,p,n)\), where \(S\) is any generating reflection set of \(G(m,p,n)\) of minimal possible cardinality and \(P_S\) is the set of all the basic relations on \(S\).
scientific article; zbMATH DE number 2136980