Decomposition numbers for unipotent blocks with small $\mathfrak{sl}_{2}$-weight in finite classical groups

Abstract

We describe a new approach to the word problem for Artin–Tits groups and, more generally, for the enveloping group \mathcal U (M) of a monoid M in which any two elements admit a greatest common divisor. The method relies on a rewrite system \mathcal R_M that extends free reduction for free groups. Here we show that, if M satisfies what we call the 3-Ore condition about common multiples, what corresponds to type FC in the case of Artin–Tits monoids, then the system \mathcal R_M is convergent. Under this assumption, we obtain a unique representation result for the elements of \mathcal U (M) , extending Ore’s theorem for groups of fractions and leading to a solution of the word problem of a new type. We also show that there exist universal shapes for the van Kampen diagrams of the words representing 1.