An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras

Abstract

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^{-}:=B^{-}\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure ${\theta }_{\mathbf{i}}^{-}:({\mathbb{C}}^{\times }{)}^{l(w_0)}\to B_{w_0}^{-}$ for each reduced word $\mathbf{i}$ of the longest element $w_0$ of Weyl group. A rational function ${\Phi }_{BK}^h={\sum }_{i\in I}{\Delta }_{w_0{\Lambda }_i,s_i{\Lambda }_i}$ on $B_{w_0}^{-}$ is called a half-potential, where ${\Delta }_{w_0{\Lambda }_i,s_i{\Lambda }_i}$ is a generalized minor. Computing ${\Phi }_{BK}^h\circ {\theta }_{\mathbf{i}}^{-}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty )$ for the finite dimensional simple Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$. In this paper, for an arbitrary reduced word $\mathbf{i}$, we give an algorithm to compute the summand ${\Delta }_{w_0{\Lambda }_i,s_i{\Lambda }_i}\circ {\theta }_{\mathbf{i}}^{-}$ of ${\Phi }_{BK}^h\circ {\theta }_{\mathbf{i}}^{-}$ in the case $i\in I$ satisfies that for any weight $\mu$ of $V(-w_0{\Lambda }_i)$ and $t\in I$, it holds $⟨h_t,\mu ⟩\in \{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type ${\mathrm{A}}_n$, ${\mathrm{B}}_n$, ${\mathrm{C}}_n$ or ${\mathrm{D}}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate ${\Phi }_{BK}^h\circ {\theta }_{\mathbf{i}}^{-}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\mathrm{G}}_2$.

41 pages