Questions tagged [representation-theory]

Representation theory is the study of mathematical objects via their symmetries. The broad idea is to study all the ways to "linearize" an object by reinterpreting its elements as a collection of matrices. This reduces problems in abstract algebra to problems in linear algebra, which are better understood, and leads to new questions in turn.

Specifically, given some algebraic object $A$, a representation of $A$ is a vector space $V$ and a a structure-preserving function $\rho \colon A \to \mathrm{End}(V)$, the ring of endomorphisms, or linear transformations, of $V$. That is, we're viewing elements of $A$ as matrices in order to "linearize" $A$ and understand its symmetries.

While some representations can tell us a great deal about the structure of $A$ in themselves, representation theorists often aim to understand all of the representations of $A$. How many different possible ways can $A$ be linearized? Usually the collection of representations of $A$ forms an abelian category. One of the most common goals in representation theory is to classify representations, or to understand the structure of the category of representations.

The usual first example that students see of representation theory is in the case of finite groups, where $A$ is a finite group, $\rho$ is a group homomorphism and $V$ is a finite-dimensional vector space over the complex numbers. However, $A$ may be a Lie group, an associative algebra, a quiver, or virtually any object with an "algebraic" structure. On the other hand, $V$ may be a vector space with coefficients in any field, a Hilbert space, a module, or virtually any object with a "linear" structure.

Modern representation theorists frequently employ tools from disparate areas of math, including category theory, harmonic analysis, homological algebra, combinatorics, and algebraic geometry. Their results are frequently useful in physics, geometry, and number theory.