Abstract: Representation theory has historically had a symbiotic relationship with physics. For example, the representation theory of the two-dimensional special unitary group is the basis of the quantum theory of angular momentum which in turn motivated Hermann Weyl's work on the representation theory of compact Lie groups. In this talk, we describe a recent manifestation of this symbiosis, namely a problem in representation theory motivated by contemporary developments in material science. In joint work with Grabovsky and Milton, we have shown that matrix algebras equipped with a suitable action of a rotation group provide the appropriate mathematical context for systematically characterizing exact relations in the study of composite materials. Extending this framework, we consider the general situation of a group acting on a finite-dimensional central simple algebra by algebra automorphisms. The fundamental problem in representation theory is to classify the subrepresentations of this algebra, but this ignores the interplay between the group action and the ring structure. A more natural question which also has physical applications is the determination of those subrepresentations which have significance in terms of the multiplicative structure of the algebra. In particular, we would like to classify the invariant subalgebras and ideals and more generally, to understand the multiplication of subrepresentations in our algebra. In this talk, we classify invariant ideals and present structure and classification theorems for invariant algebras under suitable hypotheses on the algebra. We illustrate these results in the case of compact connected Lie groups. We also discuss the general problem of the multiplication of subrepresentations when the group is simply reducible.