论文标题
Berenstein-Kazhdan装饰功能和经典谎言代数的小径的算法
An algorithm for Berenstein-Kazhdan decoration functions and trails for classical Lie algebras
论文作者
论文摘要
对于仅连接的简单连接的简单代数$ g $，众所周知，一个品种$ b_ {w_0}^ - ：= b^ - \ cap u \ overline {w_0} u $具有带正结构的几何晶体$θ^-_ {\ MATHBF {i}}：（\ Mathbb {C}^{\ times}）合理函数$φ^h_ {bk} = \ sum_ {i \ in i}δ_{w_0λ_i，s_iλ_i} $ on $ b_ {w_0}^ - $称为半功能，其中$δ__{W_0λ_I，s_i iventime}计算$φ^h_ {bk} \circθ^-_ {\ Mathbf {i}} $明确地，我们获得了$ b（\ infty）$的字符串锥或多面体实现的明确形式，用于有限的dimenite dimention dimention dimemention lie elgebra $ algebra $ \ mathfrak $ \ mathfrak， 在本文中，对于任意减少的单词$ \ mathbf {i} $，我们给出了算法来计算汇总$δ_{w_0λ_i，s_iλ_i} \ circtuminbf {\ mathbf {i}} $在I $中的情况下，$ i \满足了任何权重$ $ $ v（-w_0λ_i）$和$ t \ in i $中的任何权重，它保留$ \ langle h_t，μ\ rangle \ in \ in \ in \ {2,1,1,0，-1，1，2，2，2 \} $。特别是，如果$ \ mathfrak {g} $是类型$ {\ rm a} _n $，$ {\ rm b} _n $，$ {\ rm c} _n $或$ {\ rm d} θ^-_ {\ MathBf {i}} $。我们还将证明我们的算法在$ \ mathfrak {g} $中起作用，是类型$ {\ rm g} _2 $。
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 ${\rm G}_2$.
