An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations.

For a simply connected connected simple algebraic group G , a cell B w 0 − = B − ∩ U w 0 ‾ U is a geometric crystal with a positive structure θ i − : (C ×) l (w 0) → B w 0 −. Applying the tropicalization functor to a rational function Φ B K h = ∑ i ∈ I Δ w 0 Λ i , s i Λ i called the half decoration on B w 0 − , one can realize the crystal B (∞) in Z l (w 0). By computing Φ B K h , we get an explicit form of B (∞) in Z l (w 0). In this paper, we give an algorithm to compute Δ w 0 Λ i , s i Λ i ∘ θ i − explicitly for i ∈ I such that V (Λ i) is a minuscule representation of g = Lie (G). In particular, the algorithm works for all i ∈ I if g is of type A n. The algorithm computes a directed graph DG , called a decoration graph , whose vertices are labelled by all monomials in Δ w 0 Λ i , s i Λ i ∘ θ i − (t 1 , ⋯ , t l (w 0)). The decoration graph has some properties similar to crystal graphs of minuscule representations. We also verify that the algorithm works in some other cases, for example, the case g is of type G 2 though V (Λ i) is non-minuscule. [ABSTRACT FROM AUTHOR]