representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...

s and

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose

highest weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...

is a

fundamental weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...

. 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...

. 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 In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

, all fundamental representations are

exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...

s of the defining module. * In the case of the special unitary group SU(''n''), the ''n'' − 1 fundamental representations are the wedge products

\operatorname^k\ ^n

consisting of the

alternating tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ...

s, for ''k'' = 1, 2, ..., ''n'' − 1. * The

spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equi ...

of the twofold cover of an odd

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

, the odd

spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...

, 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 In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...

of the simple Lie group of type E₈ is a fundamental representation.

Explanation

The irreducible representations of a

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

compact

are indexed by their highest weights. These weights are the lattice points in an orthant ''Q''₊ in the

weight lattice In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...

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 In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...

, 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. See the proof of Proposition 6.17 in the case of SU(3)

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

* * {{citation, first=Brian C., last=Hall, title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, edition= 2nd, series=Graduate Texts in Mathematics, volume=222 , publisher=Springer, year=2015, isbn=978-0-387-40122-5. ;Specific Lie groups Representation theory