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arXiv 1808.03066math.GRmath.CAmath.COmath.CVmath.GT
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关于Artin-Tits幺半群的增长性和偏θ函数

On the growth of Artin--Tits monoids and the partial theta function

Ramón Flores, Juan González-Meneses

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36 pages, 1 figure
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摘要：

本文给出了一个确定齐次Garside幺半群关于原子生成集的增长函数的新方法。特别是，我们提出了一个公式的增长函数的每个Artin-山雀幺半群的球形类型（因此，每个辫子幺半群）相对于标准的发电机，作为一个非常简单的矩阵的行列式的逆。使用这种方法，我们表明，指数增长率的Artin-山雀幺半群的类型$A_n$（积极的辫子幺半群）趋于$3.233636\ldots$作为$n$趋于无穷大。这个数字是众所周知的，因为它是唯一的解决方案$x_0(y)=-(1+y+2y^2+4y^3+9y^4+\cdots)$的经典部分θ函数的系数的增长率。我们还描述了序列$1,1,2,4,9,\ldots$形成的系数的$-x_0(y)$，通过显示，其$k$ th项（系数的$y^k$）是等于辫子的长度$k$的数量，在正辫幺半群$A_{\infty}$上的无限数量的股，其最大的字典的代表开始于第一个生成元$a_1$。这是部分θ函数和辫子理论之间的意外联系。

英文摘要：

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin--Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix. Using this approach, we show that the exponential growth rates of the Artin--Tits monoids of type $A_n$ (positive braid monoids) tend to $3.233636\ldots$ as $n$ tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only solution $x_0(y)=-(1+y+2y^2+4y^3+9y^4+\cdots)$ to the classical partial theta function. We also describe the sequence $1,1,2,4,9,\ldots$ formed by the coefficients of $-x_0(y)$, by showing that its $k$th term (the coefficient of $y^k$) is equal to the number of braids of length $k$, in the positive braid monoid $A_{\infty}$ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator $a_1$. This is an unexpected connection between the partial theta function and the theory of braids.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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