dimension for each non-negative integer . For example, consider real three-dimensional space, . The group of 3D rotations SO(3) acts naturally on this space as a group of matrices. This explicit realisation of the rotation group is known as the fundamental representation , so is a representation space. The full data of the representation is . Since the dimension of this representation space is 3, this is known as the triplet representation for , and it is common to denote this as . Application to theoretical physics For applications to theoretical physics, we can restrict our attention to the representation theory of a handful of physically important groups. Many of these have well understood representation theory: : Part of the gauge group of the Standard model, and the gauge group for theories of electromagnetism. Irreps are all 1 dimensional and are indexed by integers , given explicitly by . The index can be understood as the winding number of the map. : Part of the gauge group of the Standard model. Irreps are indexed by non-negative integers in , with describing the dimension of the representation, or, with appropriate normalisation, the highest weight of the representation. In physics it is common convention to label these by half-integers instead. See Representation theory of SU(2). : The group of rotations of 3D space. Irreps are the odd-dimensional irreps of : Part of the gauge group of the Standard model. Irreps are indexed pairs of non-negative integers , describing the highest weight of the representation. See Clebsch-Gordan coefficients for SU(3). : The Lorentz group, the linear symmetries of flat spacetime. All representations arise as representations of its corresponding spin group. See Representation theory of the Lorentz group. : The spin group of . Irreps are indexed by pairs of non-negative integers , indexing the dimension of the representation. : The Poincaré group of isometries of flat spacetime. This can be understood in terms of the representation theory of the groups above. See Wigner's classification. These groups all appear in the theory of the Standard model. For theories which extend these symmetries, the representation theory of some other groups might be considered: Conformal symmetry: For pseudo-Euclidean space, symmetries are described by the conformal group . Supersymmetry: Symmetry described by a supergroup. Grand unified theories: Gauge groups which contain the Standard model gauge group as a subgroup. Proposed candidates include and . Physics Quantum field