부산대학교 도서관

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$|⁠. [ABSTRACT FROM AUTHOR]