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On the growth of Artin-Tits monoids and the partial theta function. (English) Zbl 1495.20054

A Garside monoid is a cancellative monoid where greatest common divisors and least common multiples exist and some finiteness conditions are satisfied.
The authors propose a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. They 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. They 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 formal power series \(x_0(y) =-(1+ y+2y^2+4y^3+9y^4+\cdots )\) which is the leading root of the classical partial theta function \(\sum _{k=0}^{\infty}y^{\binom{k}{2}}x^k\). They 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.

MSC:

20M32 Algebraic monoids
20F36 Braid groups; Artin groups
11F27 Theta series; Weil representation; theta correspondences
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