The Standard Model of particle physics may seem complicated and arbitrary,
but it has hidden patterns that are revealed by the relationship between three
"grand unified theories": theories that unify forces and particles by extending
the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These
three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on
the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x
SU(4). In this expository account for mathematicians, we explain only the
portion of these theories that involves finite-dimensional group
representations. This allows us to reduce the prerequisites to a bare minimum
while still giving a taste of the profound puzzles that physicists are
struggling to solve.Comment: 73 pages, 20 ps figure