Representation theory is a field of Mathematics, known in 1896 by the German mathematician Frobenius, that has lots of applications in physics, number theory, and cryptography.
In representation theory, mathematicians study representations of algebras (group, rings, topological spaces) by representing their elements as linear transformations of vector spaces. More specifically, a representation makes abstract algebraic objects more concrete by transforming them into matrices.
The importance of representation theory is that abstract problems are reduced to problems in linear algebra which is a well-known theory in mathematics.
An important branch of representation theory is "group
representation theory", in which the group elements are represented by invertible matrices. Consequently, we work with matrix multiplication instead of working with the original group operation.
More exactly, a "representation" of group G means a homomorphism mapping from G to the automorphism group of the object.
This asks for a possible way to view a group as a permutation group or a linear group. More narrowly, it considers homomorphisms from the group into the matrix group GLn(C), where C is most frequently the field of complex numbers.
Also, representation theory depends heavily on the type of vector space on which the algebra acts. One must consider the type of field over which the vector space is defined. The most important cases are the field of complex numbers and the field of real numbers.