Growth of braid monoids and the partial theta function

10 enero 2019: Seminario de Análisis, 18:00-19:00

Conferenciante: Juan González-Meneses (Universidad de Sevilla e IMUS)

Título: Growth of braid monoids and the partial theta function

Resumen: 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 Σ y^{k(k-1)/2} x^k.

We also describe the sequence 1,1,2,4,9,... 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_∞ 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 (joint with Ramón J. Flores).