Some Shallow Elements of Coxeter Groups of Type B

Annual Meeting in Mathematics 2024

Abstract

The length and reflection length are two fundamental statistics on an element of a Coxeter group. In 2015, Petersen and Tenner [T.K. Petersen, B.E. Tenner, The depth of a permutation, J. Comb. 6 (1-2) (2015) 145-178] introduced a new statistic called depth. In the case of Coxeter groups of type A, Woo [A. Woo, The shallow permutations are the unlinked permutations, arXiv preprint (2022) arXiv:2201.12949] provided a characterization of shallow elements in terms of their cycle diagrams. In this paper, we introduce cycle diagrams associated with Coxeter groups of type B. Furthermore, we characterize certain shallow elements in Coxeter groups of type B using their cycle diagrams, while we provide a necessary condition for other elements.