# Fundamental representation

In representation theory of Lie groups and Lie algebras, a **fundamental representation** is an irreducible finite-dimensional representation of a semisimple Lie group
or Lie algebra whose highest weight is a fundamental weight. 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. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weights. These weights are the lattice points in an orthant *Q*_{+} in the weight lattice of the Lie group consisting of the dominant integral 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 combinations of the fundamental weights.^{[1]} The corresponding irreducible representations are the **fundamental representations** of the Lie group. From the expansion of a dominant weight in terms of the fundamental weights one can take a corresponding tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight.^{[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).