Seminario de teoría de grupos UAM-ICMAT

Ponente: Juan González-Meneses (Universidad de Sevilla)

Title: Growth of braid monoids and the partial theta function

Miercoles  10/10/2018, 11:30, Aula 520, UAM

Abstract: Joint with Ramón J. Flores.

We present a new procedure to determine the growth function of an Artin-Tits monoid of spherical type (hence of a 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 positive braid monoids $A_n$  tend to 3.233636… as $n$ tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only solution $x_0(y)=−(1+y+2y^{^2}+4y^{^3}+9y^{^4}+⋯)$ to the classical partial theta function $sum_{k=0}^{infty}{y^{kchoose 2} x^k}$.

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.