经典李代数的Berenstein-Kazhdan装饰函数和迹的算法
An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras
摘要：
对于单连通单代数群$G$，已知一个簇$B_{w_0}^-:=B^-\cap U\overline{w_0}U$对Weyl群的最长元$w_0$的每一个约化字$\mathbf{i}$都具有正结构$θ^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$的几何晶体结构. $B_{w_0}^-$上的有理函数$Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$称为半势，其中$Δ_{w_0Λ_i,s_iΛ_i}$是广义子式。通过显式计算$Φ^h_{BK}\circ θ^-_{\mathbf{i}}$，得到了有限维单李代数$\mathfrak{g}={\rm Lie}(G)$的$B(\infty)$的弦锥或多面体实现的显式形式.本文对任意约简字$\mathbf{i}$，给出了一个计算$Φ^h_{BK}\circ θ^-_{\mathbf{i}}$的和项$Δ_{w_0Λ_i,s_iΛ_i}\circ θ^-_{\mathbf{i}}$的算法，条件是$i\in I$满足对$V(-w_0Λ_i)$和$t\in I$的任意权$μ$，它保持$\langle h_t,μ\rangle\in\{2,1,0,-1,-2\}$ .特别地，如果$\mathfrak{g}$是${\rm A}_n$、${\rm B}_n$、${\rm C}_n$或${\rm D}_n$类型，则所有$i\in I$都满足该条件，从而可以完全计算出$Φ^h_{BK}\circ θ^-_{\mathbf{i}}$。我们还将证明，我们的算法工作的情况下，$\mathfrak{g}$是类型${\rm G}_2$。
英文摘要：
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 $θ^-_{\mathbf{i}}:(\mathbb{C}^{\times})^{l(w_0)}\rightarrow B_{w_0}^-$ for each reduced word $\mathbf{i}$ of the longest element $w_0$ of Weyl group. A rational function $Φ^h_{BK}=\sum_{i\in I}Δ_{w_0Λ_i,s_iΛ_i}$ on $B_{w_0}^-$ is called a half-potential, where $Δ_{w_0Λ_i,s_iΛ_i}$ is a generalized minor. Computing $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ 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 $Δ_{w_0Λ_i,s_iΛ_i}\circ θ^-_{\mathbf{i}}$ of $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$ in the case $i\in I$ satisfies that for any weight $μ$ of $V(-w_0Λ_i)$ and $t\in I$, it holds $\langle h_t,μ\rangle\in\{2,1,0,-1,-2\}$. 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 $Φ^h_{BK}\circ θ^-_{\mathbf{i}}$. We will also prove that our algorithm works in the case $\mathfrak{g}$ is of type ${\rm G}_2$.
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