Singularities of Schubert Varieties within a Right Cell

@article{Lanini2021SingularitiesOS,
  title={Singularities of Schubert Varieties within a Right Cell},
  author={Martina Lanini and Peter J. McNamara},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2021}
}
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… 
Categorical diagonalization and $p$-cells
In the Iwahori-Hecke algebra, the full twist acts on cell modules by a scalar, and the half twist acts by a scalar and an involution. A categorification of this statement, describing the action of
Cellularity of the p-Canonical Basis for Symmetric Groups.
For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig

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