Fundamental representation

Short description: Aspect of mathematical representation theory

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

• In the case of the general linear group, all fundamental representations are 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 ${\mathrm{Alt}}^k{\mathbb{ℂ}}^n$ 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 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 combination 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

In mathematical physics, the term fundamental representation is also used to refer to the defining representation of a matrix Lie group or matrix Lie algebra, that is, its standard representation as a subgroup of $\mathrm{GL}(n,\mathbb{ℂ})$ or subalgebra of $\mathfrak{𝔤}\mathfrak{𝔩}(n,\mathbb{ℂ})$. The complex conjugate of the fundamental representation is known as the antifundamental (or anti-fundamental) representation. Although the distinction between the fundamental and the antifundamental representation is a matter of convention, these two are often non-equivalent, because each of them is a complex representation.

References

• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9. 
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-0-387-40122-5 .

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