Highest weight modules play a key role in the representation theory of several classes of algebraic objects occurring in Lie theory, including Lie algebras, Lie groups, algebraic groups, Chevalley groups and quantized enveloping algebras. In many of the most important situations, the weights may be regarded as points in Euclidean space, ℝn, and there is a finite group (called a Weyl group) that acts on the set of weights by linear transformations. The minuscule representations are those for which the Weyl group acts transitively on the weights, and the highest weight of such a representation is called a minuscule weight. The term “minuscule weight” is a translation of Bourbaki's term poids minuscule [8, VIII, section 7.3]; the spelling “miniscule” is also found in the literature, although less commonly, and Russian-speaking authors often call minuscule weights microweights. The list of minuscule representations is as follows: all fundamental representations in type An, the natural representations in types Cn and Dn, the spin representations in types Bn and Dn, the two 27-dimensional representations in type E6 and the 56-dimensional representation in type E7.

Minuscule weights and minuscule representations are important because they occur in a wide variety of contexts in mathematics and physics, especially in representation theory and algebraic geometry. Minuscule representations are the starting point of Standard Monomial Theory developed by Lakshmibai, Seshadri and others [42], and they play a key role in the geometry of Schubert varieties.