  We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any
Bruhat interval of a symmetric group into an interval whose extremes lie in the
same right Kazhdan-Lusztig cell. This apparently harmless fact has applications
in finding examples of reducible associated varieties of
$\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs
for symmetric groups, and in comparing various bases of irreducible
representations of the symmetric group or its Hecke algebra. For example, we
are able to systematically produce many negative answers to a question from the
1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via
computer calculations only in 2014.
