In [Journal of Pure and Applied Algebra 224 (2020), no 12, 106449], V. Mazorchuk and R. Mrđen (with some help by A. Hultman) prove that, given a Weyl group, the intersection of a Bruhat interval with a parabolic coset has a unique maximal element and a unique minimal element. We show that such intersections are actually Bruhat intervals also in the case of an arbitrary Coxeter group.
Bruhat intervals and parabolic cosets in arbitrary Coxeter groups / Marietti, M.. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - STAMPA. - 614:(2023), pp. 1-4. [10.1016/j.jalgebra.2022.09.023]