Fundamental representation

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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.

Examples[edit]

  • In the case of the general linear group, all fundamental representations are exterior powers of the defining module.
  • In the case of the special unitary group SU(n), the n − 1 fundamental representations are the wedge products \operatorname{Alt}^k\ {\mathbb C}^n consisting of the alternating tensors, for k = 1, 2, ..., n − 1.
  • The spin representation of the twofold cover of an odd orthogonal group, the odd spin group, and the two half-spin representations of the twofold cover of an even orthogonal group, the even spinor group, are fundamental representations that cannot be realized in the space of tensors.
  • The adjoint representation of the simple Lie group of type E8 is a fundamental representation.

Explanation[edit]

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. 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.

Other uses[edit]

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).

References[edit]

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6