BOOK REVIEW Combinatorics of minuscule representations (Cambridge Tracts in Mathematics 199)

To be a minuscule representation of a complex simple Lie algebra g is to be ‘as small as possible’: not only irreducible and finite-dimensional, but with all weights in the same orbit under the action of the Weyl group W. In particular, the highest weight λ has one-dimensional weight space, so all of its weight spaces are one-dimensional. It turns out to be fruitful to partially order the set of weights Wλ, defining the weight poset in which μ ν when ν − μ is a non-negative sum of positive roots. As explained in R. M. Green’s book under review here, poset structures are the key to a minuscule representation’s many special properties, making it simpler than a typical g-irreducible. Minuscule representations exist only in types A,B,C,D,E6, E7. In type A, they are the exterior powers ∧kCn of the natural representation Cn for the Lie algebra g: = sln(C) of trace zero matrices in Cn×n. Here the Weyl group W of type An−1 can be taken to be the symmetric group Sn permuting the standard basis e1,..., en. The monomial wedges {ei1 ∧ · · · ∧ eik}1i1<···<ikn give a basis of weight vectors for ∧kCn, depicted here for ∧2C5, ordered as in the weight poset: e1 ∧ e2 e1 ∧ e3 e2 ∧ e3 e1 ∧ e4 e2 ∧ e4 e1 ∧ e5 e3 ∧ e4 e2 ∧ e5 e3 ∧ e5 e4 ∧ e