For example, the defining module of a classical Lie group is a fundamental representation.
Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan.
The irreducible representations of a simply-connected compact Lie group are indexed by their highest weights.
It can be proved that there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram, such that any dominant integral weight is a non-negative integer linear combination of the fundamental weights.
[2] Outside of Lie theory, the term fundamental representation is sometimes loosely used to refer to a smallest-dimensional faithful representation, though this is also often called the standard or defining representation (a term referring more to the history, rather than having a well-defined mathematical meaning).