Garside Families in Artin–tits Monoids and Low Elements in Coxeter Groups

where both terms consist of two alternating letters and have the same length. First investigated by J.Tits in the late 1960s [2], and then in [3] and [11], these groups remain incompletely understood, with many open questions, including the decidability of the Word Problem in the general case [6]. The only well understood case is the one of spherical type, which is the case when the associated Coxeter group, obtained by adding the relations s = 1 to the presentation, is finite. Then a large part of the known results in this case is included in the fact that an Artin– Tits group of spherical type is a Garside group, and the corresponding monoid is a Garside monoid [10, 7]. At the heart of the properties of an Artin–Tits monoid of spherical type—and more generally of a Garside monoid—lies the fact that every element of the latter admits a distinguished decomposition (“greedy normal form”) involving the divisors of a certain element ∆ (“Garside element”), in which each entry is in a sense maximal [17, Chapter 9]. It was recently realized that such distinguished decompositions exist in the more general framework of what was called Garside families: whenever F is a Garside family in a left-cancellative monoid M (or category), the mechanism of the greedy normal form works and provides distinguished decompositions with nice properties [8, 9]. The case of a Garside monoid corresponds to a Garside family consisting of the divisors of a single element ∆ (“bounded Garside family”), but various examples of unbounded Garside families are now known. If M is an Artin–Tits monoid of non-spherical type, that is, the associated Coxeter group W is infinite, it is well known that M is not a Garside monoid: the projection of a possible Garside element to W should be a longest element of W , which cannot exist in this case. This however says nothing about possible