Tensor product of representations

The tensor product of representations combines two mathematical ways (representations) in which a group acts on spaces, creating a new, more complex action on a combined space. Think of each representation as a way a group influences, say, two different objects or systems. The tensor product merges these influences into a single, unified action that captures interactions between them, providing insight into how combined systems behave under group symmetries. This operation helps mathematicians analyze complex symmetries by building larger representations from simpler ones, much like combining tools to understand more intricate structures.