Weyl group actions on minuscule weights in Maple.

The purpose of this worksheet is to explore the action of Weyl groups (associated with simple Lie algebras) on weights of representations (particularly minuscule representations). Specifically, we want to know what kind of cycle structures appear when we view the Weyl group's action as a permutations of weights.

In general, a representation's weights may split up into several orbits. This can happen even when the representation is irreducible. In fact, since the Weyl group action preserves inner products (and thus preserves lengths), we must have multiple orbits anytime we have a non-trivial representation with the zero vector as a weight. However, in the case of minuscule representations, all weights lie in a single orbit.

A minuscule representation is an irreducible representation in which all the weights lie in a single Weyl group orbit. The highest weight of such a representation is said to be a minuscule weight.

We know that the highest weight space is always one-dimensional. Also, if two weights are in the same Weyl group orbit, then their weight spaces must have the same dimensions. Thus, all of the weight spaces of a minuscule representation are one-dimensional.

Thus, minuscule representations behave quite nicely with respect to the Weyl group action. In fact in some cases we can "see" the irreducibility of the minuscule representation by inspecting the cycle structures of the permutations associated with the Weyl group's elements. An invariant subspace of a representation cannot share part of an orbit of weights, so if the cycle structures tell us that there is only one orbit, we must conclude that the representation is irreducible (this assumes that the weight spaces are one-dimensional, which may fail to hold if the highest weight is not minuscule).

In this worksheet, we will see that cycle structure determines irreducibility for minuscule representation associated with simple Lie algebras of types B2, B3, B5, B7, E6, and E7. The corresponding paper, On the Constructive Inverse Problem in Differential Galois Theory, covers types An, Cn, and Dn (all n). We will also see that cycle structure does not tell us enough in the case of B4 -- even though we are dealing with an irreducible minuscule representation.

The first link gives a Maple worksheet that should run under Maple 7 or Maple 8. The second link gives a worksheet for Maple 9. The final link is a static version (exported to html) of a Maple session using the worksheet. Calculations mentioned in the above paper concerning simple Lie algebras of types B2, B3, B4, B5, B7, E6, and E7 appear here.