Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 070, 9 pages      arXiv:2003.08616      https://doi.org/10.3842/SIGMA.2021.070

Singularities of Schubert Varieties within a Right Cell

Martina Lanini a and Peter J. McNamara b
a) Department of Mathematics, University of Rome ''Tor Vergata'', Italy
b) School of Mathematics and Statistics, The University of Melbourne, Australia

Received December 10, 2020, in final form July 06, 2021; Published online July 19, 2021

Abstract
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.

Key words: Schubert varieties; interval pattern embedding; Kazhdan-Lusztig cells; Specht modules.

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