Computational Abstract Algebra: Using Monomial Matrices to Represent Groups in GAP

Abstract

A monomial matrix is a matrix with exactly one non-zero element in each row and column. We will utilize GAP to construct all (transitive) representations of a given group using monomial matrices. First, essential group theory definitions and theorems will be provided, as well as an in-depth look at table of marks and monomial matrices. After describing the necessary mathematics, we will explore the GAP programming needed to achieve this goal. Ultimately we want a table, where each row represents a subgroup of the given group and, within the row, the table will hold the linear characters fixed by the monomial matrices of that subgroup. We will furthermore explore how to represent monomial matrices computationally in different ways and how to create a data structure to represent them. Our final goal will require GAP functions for finding all homomorphisms from a subgroup to roots of unity, using these homomorphisms to create monomial matrix representations of the group, and iterating through the subgroups of the group (up to conjugacy) to find all (transitive) monomial matrix representations of the group.