Representation theory is a way of taking complicated objects and “representing” them with simpler objects. The complicated objects are often collections of mathematical objects — like numbers or symmetries — that stand in a particular structured relationship with each other. These collections are called groups. The simpler objects are arrays of numbers called matrices, the core element of linear algebra. While groups are abstract and often difficult to get a handle on, matrices and linear algebra are elementary.
A representation provides a simplified picture of a group, just as a grayscale photo can serve as a low-cost imitation of the original color image. Put another way, it “remembers” some basic but essential information about the group while forgetting the rest. Mathematicians aim to avoid grappling with the full complexity of a group; instead they gain a sense of its properties by looking at how it behaves when converted into the stripped-down format of linear transformations.
“We don’t have to look at the group at once,” Norton said. “We can look at a representation that’s smaller and still understand something about our group.”
A group can almost always be represented in multiple ways. S3, for example, has three distinct representations when real numbers are used to fill in the matrices: the trivial representation, the reflection representation and the sign representation (emphasis added).