Ultra-Low elements and Join Irreducible gates in Coxeter groups

Abstract: In this note, we study the "tight gates" of some "regular" partitions in Coxeter groups. In particular, we study the tight gates of the cone type partition of a Coxeter group $W$; a partition giving rise to the minimal automaton recognising the language of reduced words of $W$ recently studied by Parkinson and Yau (2022) and Przytycki and Yau (2024). We show that "tight gates" are the join-irreducible elements of the $m$-low Garside shadows for $m \in \mathbb{N}$ and the smallest Garside shadow of $W$ under the right weak order. In the case of the tight gates of the cone type partition, we show that they also act as a "gate" of the witnesses of boundary roots of cone types. As an application of these results, we give a very efficient method of determining whether two elements have the same cone type without computing the minimal automaton. We also apply our results to explicitly describe the set of ultra-low elements in some select classes of Coxeter groups, thereby verifying a conjecture of Parkinson and Yau for these groups. Furthermore, we explicitly describe the tight gates of the cone type partition in these cases.