weight vector in a representation
weight of a weight vector
applying a raising/lowering operator
tensoring two weight vectors
irreducibility of a representation
path in t^*
endpoint of the path
applying a "path operator", to change the endpoint
concatenating two paths
connectedness of the colored graph of paths
It is easiest to define the path operators infinitesimally, and then say "take the time one flow".
To lower a path in a particular direction X, find the point on it with greatest dot product in that direction. There may be several; break ties by using the last such point, i.e. latest as traced from the origin. Past that point the path goes more in the direction -X. Take a little segment of that part and reflect it through the hyperplane perpendicular to X, leaving the rest of the path alone. The effect will be to nudge the latter part of the path in the direction X.
The subtlety in doing following the time one flow of this operation is that the hot spot can move; after a while the surgery on the path takes place near a different point. In some sense the operators are continuous but not differentiable.
To lower a path is the inverse operation. One looks for the point with the least dot product, and the ties are broken the opposite way.
If the hot spot is the endpoint of the path, the path operator is declared to be inapplicable (Littelmann says it produces "0", which is in some basically not a path). There is an annoying subtlety; the infinitesimal operators may be applicable while the finite ones (the time one flows) are not. This will be resolved in the next section.
The first test program lets one play with a path and the lowering operators. Someday soon I will implement honest raising operators!
Almost the same story works in the path model, except that a dominant path is not one for which all raising operators are inapplicable, but one for which all infinitesimal raising operators are inapplicable. This is the same as its being in the positive Weyl chamber.
The second test program lets one choose an initial path (Warning: at present it trusts you to choose a dominant path) and lowers it to produce all other paths in the representation.
The number of paths with a given endpoint nu is the dimension of the mu weight space in the representation lambda.
The colored graphs do not depend on the original path, just the weight; they are Kashiwara's crystal graphs.
Take all concatenations of a lambda-path and a mu-path. This set will also be closed under the root operators, and so form its own colored graph - but will not be connected. It has a component for each irreducible subrepresentation of the tensor product, and in each component is a unique dominant path. Such paths are necessarily the original path with endpoint lambda with a mu-path on the end, such that the concatenation is dominant.
A plactic algebra for semisimple Lie algebras. Adv. Math. 124 (1996), no. 2, 312--331. 17Bxx
The path model for representations of symmetrizable Kac-Moody algebras. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 298--308, Birkhduser, Basel, 1995. 17Bxx
Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499--525.
Crystal graphs and Young tableaux. J. Algebra 175 (1995), no. 1, 65--87.
A Littlewood-Richardson rule for
symmetrizable Kac-Moody algebras. Invent. Math. 116 (1994), no. 1-3,
The first test program allows you to apply the operators yourself to a path. Commands:Note: the right-moving operator should really be a raising operator, which means it takes the first-moving opportunity to raise rather than the last. So mathematically it's not so correct to include it.
To specify a path, move the mouse within the positive Weyl chamber, and click on turning points, including the endpoint. Then hit space. The endpoint will be moved to a nearby weight.
If you have it running freely, and your computer isn't quick enough to keep up with its drawing the paths, type "a" to tell it to spend more time drawing, until you can see the paths. If you want it to run faster type "A".
Coming soon: other root systems, tensor products. What else do you want? Send me mail.