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

. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.

Examples

• In the case of the
  exterior products
  of the defining module.
• In the case of the special unitary group
  SU(n)
  , the n − 1 fundamental representations are the wedge products ${\displaystyle \operatorname {Alt} ^{k}\ {\mathbb {C} }^{n}}$ 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

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 combination 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]

Other uses

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

• OCLC 246650103
  .
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
  ISBN 978-0-387-40122-5
  .

Specific