Minuscule representation

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.

• A_n Dimension = (n+1
  k) for 0 ≤ k ≤ n (exterior powers of vector representation)
• B_n Dimension 1 (trivial), dimension 2ⁿ (spin)
• C_n Dimension 1 (trivial), dimension 2n (vector)
• D_n 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)

References

• Seshadri, C. S. (1978), "Geometry of G/P. I. Theory of standard monomials for minuscule representations", C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., 8, Berlin, New York: Springer-Verlag, pp. 207–239