Last update: 1 December 2012
A highest weight path is a path such that
If is a highest weight path with then, necessarily, . The following theorem gives an expression for the character of a crystal in terms of the basis of .
Let be a crystal. Let be as defined in (5.24) and as in (5.25). Then
be the element of the
-string of which satisfies
Then and Hence .
Since , and
There is some cancellation which can occur in this sum. Assume such that and let be the first time that leaves the cone . In other words, let be minimal such that there exists an with
Let be the minimal index such that the point and define to be the element of the -string of such that
Note that since is not the head of its -string and is well defined. If then the first time that leaves the cone is the same as the first time that leaves the cone and . Thus and . Since the terms and cancel in the sum in (5.29). Thus
Recall the notations for Weyl characters, tensor product multiplicities, restriction multiplicities and paths from (5.5), (5.11), (5.17), (5.22). For each fix a highest weight path with endpoint and let Let and let Then
(a) The path is the unique ighest weight path in . Thus, by Theorem 5.5,
(b) By the "Leibnitz formula" for the root operators in Theorem 5.8c, the set is a crystal. Since where the third equality is from Theorem 5.5 and the last equality is because the path has and is the only highest weight path in
(c) A -crystal is a set of paths which is closed under the operators , for . Since the statement applies by applying Theorem 5.5 to viewed as a -crystal.
These notes are intended to supplement various lecture series given by Arun Ram. This is a typed version of section 5.4 of the paper [Ram2006].
[Ram2006] Alcove walks, Hecke algebras, Spherical functions, crystals and column strict tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (Special Issue: In honor of Robert MacPherson, Part 2 of 3) (2006) 963-1013. MR2282411