An Algorithm for Berenstein–Kazhdan Decoration Functions and Trails for Classical Lie Algebras

Abstract

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