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 $\theta^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\mathbf{i}$ 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 $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$ explicitly, we get an explicit form of string cone or polyhedral realization of $B(\infty)$ for the finite dimensional simple Lie algebra $\mathfrak{g}={\rm Lie}(G)$. 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 $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$ in the case $i\in I$ satisfies that for any weight $\mu$ of $V(-w_0\Lambda_i)$ and $t\in I$, it holds $\langle h_t,\mu \rangle\in\{2,1,0,-1,-2\}$. In particular, if $\mathfrak{g}$ is of type ${\rm A}_n$, ${\rm B}_n$, ${\rm C}_n$ or ${\rm D}_n$ then all $i\in I$ satisfy this condition so that one can completely calculate $\Phi^h_{BK}\circ \theta^-_{\mathbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\rm G}_2$.