First, please pardon the ignorance behind this question. I know a fair amount of math but almost no physics.
I'm hoping someone can give me a brief "big picture" explanation of how physicists were able to predict the existence of the Higgs boson. Here's a (perhaps completely wrong) example of the level of explanation I'm hoping for:
"Consider the following group [insert precise specification of a particular group here]. This group is know to have precisely [insert number here] irreducible representations. All but one of these representations corresponds to a previously-observed particle via a correspondence in which the property [insert mathematical property of representation here] corresponds to the property [insert physical property of particle here]. The Higgs Boson is the particle that corresponds to the remaining representation."
If I happen to have hit on the right story, I'd like someone to fill in the blanks. If, as is more likely, I've concocted a completely wrong story, I'd like someone to give me a more accurate story at about the same level of sophistication. Thanks!