Minuscule representation
Encyclopedia
In mathematical representation theory, a minuscule representation of a semisimple Lie algebra or group is an irreducible representation such that the Weyl group
acts transitively on the weights.
The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the minuscule representations are given as follows.
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
acts transitively on the weights.
The minuscule representations are indexed by the weight lattice modulo the root lattice, or equivalently by irreducible representations of the center of the simply connected compact group. For the simple Lie algebras, the minuscule representations are given as follows.
- An Dimension = for 0 ≤ k ≤ n (exterior powers of vector representation)
- Bn Dimension 1 (trivial), dimension 2n (spin)
- Cn Dimension 1 (trivial), dimension 2n (vector)
- Dn Dimension 1 (trivial), dimension 2n (vector), dimension 2n−1 (half spin), dimension 2n−1 (half spin).
- E6 Dimension 1 (trivial), dimension 27, dimension 27
- E7 Dimension 1 (trivial), dimension 56
- E8 Dimension 1 (trivial)
- F4 Dimension 1 (trivial)
- G2 Dimension 1 (trivial)