论文标题
树木枚举清晰度的标准和Kuperberg的G2 Spider中的三角形的渐近数
A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider
论文作者
论文摘要
我们证明了库珀贝格的渐近公式从谎言代数$ g_2 $的代表理论中。给定一个非阴性序列$（a_n）_ {n \ geq 1} $，用于生成函数的身份$ b（x）= a（xb（x））$用于生成函数$ a（x）= 1+\ sum_ {n \ geq 1} a_n x^n $和$ b（x） $ b_n $的生根平面树，带有$ n $顶点，以便每个具有$ i $ ridy的顶点都可以具有$ a_i $不同的颜色之一。在\ cite {kuperberg}中证明了库珀伯格在$ b_n = \ dim \ text {inv} _ {g_2}（v（λ_1）^{\λ_1）^{\ otimes n}）$中，其中$ v（λ_1）$是7二维的$ g a_n是$ g的$ g_2常规$ n $ gon的三角剖分，使每个内部顶点的学位至少为$ 6 $。他还观察到，$ \ limsup_ {n \ to \ infty} \ sqrt [n] {a_n} \ leq 7/b（1/7）$，并推测，这种估计值是尖锐的，或者在权力系列中，convergence of power servius of power servius of ctripgence of clentgence of $ a（x）$ a（x）$ a（x）$ a $ b（1/7 $ 7 $ 7 $/7 $ 7 $/7 $ $ $ a（x）。我们通过在General Power Series $ a（x）$ A（x）$和满足$ b（x）= a（xb（x））$的类似估算中引入新的清晰度标准来证明这一猜想。此外，通过在$ b（x）$的近期发现的生成函数上进行的奇异性分析，我们通过得出序列$（a_n）$的渐近公式来显着完善猜想。
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 $(a_n)$.
