树计数中的锐度准则与Kuperberg的G₂蜘蛛中的三角剖分渐近数量
A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider
AI总结：
本文证明了Kuperberg关于李代数$G_2$表示论的三角剖分渐近公式猜想，提出树计数锐度准则并通过奇点分析细化了正则$n$边形三角剖分数量的渐近估计。
摘要：
我们证明了Kuperberg提出的关于李代数$G_2$表示论的一个渐近公式猜想。给定非负序列$(a_n)_{n\geq 1}$，生成函数$A(x)=1+\sum_{n\geq 1} a_n x^n$和$B(x)=1+\sum_{n\geq 1} b_n x^n$满足恒等式$B(x)=A(xB(x))$，该恒等式确定了具有$n$个顶点的根平面树的数量$b_n$，其中每个有$i$个孩子的顶点可具有$a_i$种不同颜色。Kuperberg在文献[Kuperberg]中证明，当$b_n=\dim \text{Inv}_{G_2} (V(λ_1)^{\otimes n})$（其中$V(λ_1)$是$G_2$的7维基本表示）且$a_n$是正则$n$边形的三角剖分数量（每个内部顶点度数至少为6）时，该恒等式成立。他还观察到$\limsup_{n\to\infty}\sqrt[n]{a_n}\leq 7/B(1/7)$，并猜想此估计是锐的，即幂级数$A(x)$的收敛半径恰好为$B(1/7)/7$。我们通过引入一般幂级数$A(x)$和$B(x)$满足$B(x)=A(xB(x))$时类似估计的新锐度准则来证明该猜想。此外，通过对最近发现的$B(x)$生成函数进行奇点分析，我们推导了序列$(a_n)$的渐近公式，显著细化了该猜想。
英文摘要：
We prove a conjectured asymptotic formula of Kuperberg from the representation theory of the Lie algebra $G_2$. Given a non-negative sequence $(a_n)_{n\geq 1}$, the identity $B(x)=A(xB(x))$ for generating functions $A(x)=1+\sum_{n\geq 1} a_n x^n$ and $B(x)=1+\sum_{n\geq 1} b_n x^n$ determines the number $b_n$ of rooted planar trees with $n$ vertices such that each vertex having $i$ children can have one of $a_i$ distinct colors. Kuperberg proved in \cite{Kuperberg} that this identity holds in the case that $b_n=\dim \text{Inv}_{G_2} (V(λ_1)^{\otimes n})$, where $V(λ_1)$ is the 7-dimensional fundamental representation of $G_2$, and $a_n$ is the number of triangulations of a regular $n$-gon such that each internal vertex has degree at least $6$. He also observed that $\limsup_{n\to\infty}\sqrt[n]{a_n}\leq 7/B(1/7)$ and conjectured that this estimate is sharp, or in terms of power series, that the radius of convergence of $A(x)$ is exactly $B(1/7)/7$. We prove this conjecture by introducing a new criterion for sharpness in the analogous estimate for general power series $A(x)$ and $B(x)$ satisfying $B(x)=A(xB(x))$. Moreover, by way of singularity analysis performed on a recently-discovered generating function for $B(x)$, we significantly refine the conjecture by deriving an asymptotic formula for the sequence $(a_n)$.
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