The path model 12 . 01 . 05

Let λ ∈ P . The straight line path to λ is the map pλ : [0, 1]→ hR given by pλ(t) = λt. Let `1, `2 ∈ R≥0. The concatenation of maps p1 : [0, `1] → hR and p2 : [0, 1] → hR is the map p1 ⊗ p2 : [0, `1 + `2]→ hR given by (p1 ⊗ p2)(t) = { p1(t), for t ∈ [0, `1], p1(`1) + p2(t− `1), for t ∈ [`1, `1 + `2]. Let r, ` ∈ R≥0. The r-stretch of a map p : [0, `]→ hR is the map rp : [0, `]→ hR given by (rp)(t) = r · p(t/r). The reverse of a map p : [0, `]→ hR is the map p∗ : [0, `]→ hR given by p∗(t) = p(`− t)− p(`). The weight of a map p : [0, `]→ hR is the endpoint of p, wt(p) = p(`). Let Buniv be the set of maps generated by the straight line paths under concatenation, reversing and stretching. A path is an element of Buniv. A crystal is a set of paths B that is closed under the action of the root operators ẽi : Buniv −→ Buniv ∪ {0} and f̃i : Buniv −→ Buniv ∪ {0}, 1 ≤ i ≤ n, which are defined and constructed below, in Theorem ??? and in ???, respectively. Let B be a set of paths (a subset of Buniv). The crystal graph of B is the graph with vertices B edges p′ i ←−p if p′ = f̃ip. The character of B is char(B) = ∑