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

SIGMA 17 (2021), 070, 9 pages      arXiv:2003.08616

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

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.

pdf (373 kb)   tex (19 kb)  


  1. Abe H., Billey S., Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, in Schubert Calculus - Osaka 2012, Adv. Stud. Pure Math., Vol. 71, Editors H. Naruse, T. Ikeda, M. Masuda, T. Tanisaki, Math. Soc. Japan, Tokyo, 2016, 1-52, arXiv:1403.4345.
  2. Ariki S., Robinson-Schensted correspondence and left cells, in Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math., Vol. 28, Kinokuniya, Tokyo, 2000, 1-20, arXiv:math.QA/9910117.
  3. Borho W., Brylinski J.-L., Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math. 80 (1985), 1-68.
  4. Bosma W., Cannon J., Playoust C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235-265.
  5. Garsia A.M., McLarnan T.J., Relations between Young's natural and the Kazhdan-Lusztig representations of $S_n$, Adv. Math. 69 (1988), 32-92.
  6. Graham J.J., Lehrer G.I., Cellular algebras, Invent. Math. 123 (1996), 1-34.
  7. Howlett R., Nguyen V.M., W-graph magma programs, available at, 2013.
  8. Jensen L.T., The ABC of $p$-cells, Selecta Math. (N.S.) 26 (2020), 28, 46 pages, arXiv:1901.02323.
  9. Jensen L.T., Cellularity of the $p$-canonical basis for symmetric groups, arXiv:2009.11715.
  10. Jensen L.T., Williamson G., The $p$-canonical basis for Hecke algebras, in Categorification and Higher Representation Theory, Contemp. Math., Vol. 683, Amer. Math. Soc., Providence, RI, 2017, 333-361, arXiv:1510.01556.
  11. Joseph A., On the variety of a highest weight module, J. Algebra 88 (1984), 238-278.
  12. Kashiwara M., Saito Y., Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9-36, arXiv:q-alg/9606009.
  13. Kazhdan D., Lusztig G., Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184.
  14. Lakshmibai V., Sandhya B., Criterion for smoothness of Schubert varieties in ${\rm Sl}(n)/B$, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), 45-52.
  15. McLarnan T.J., Warrington G.S., Counterexamples to the 0-1 conjecture, Represent. Theory 7 (2003), 181-195, arXiv:math.CO/0209221.
  16. McNamara P.J., Non-perverse parity sheaves on the flag variety, arXiv:1812.00178.
  17. Nguyen V.M., Type $A$ admissible cells are Kazhdan-Lusztig, Algebr. Comb. 3 (2020), 55-105, arXiv:1807.07457.
  18. Stanley R.P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  19. Tanisaki T., Characteristic varieties of highest weight modules and primitive quotients, in Representations of Lie Groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., Vol. 14, Academic Press, Boston, MA, 1988, 1-30.
  20. Vilonen K., Williamson G., Characteristic cycles and decomposition numbers, Math. Res. Lett. 20 (2013), 359-366, arXiv:1208.1198.
  21. Williamson G., On an analogue of the James conjecture, Represent. Theory 18 (2014), 15-27, arXiv:1212.0794.
  22. Williamson G., A reducible characteristic variety in type $A$, in Representations of Reductive Groups, Progr. Math., Vol. 312, Birkhäuser/Springer, Cham, 2015, 517-532, arXiv:1405.3479.
  23. Williamson G., Schubert calculus and torsion explosion (with a joint appendix with Alex Kontorovich and Peter J. McNamara), J. Amer. Math. Soc. 30 (2017), 1023-1046, arXiv:1309.5055.
  24. Woo A., Interval pattern avoidance for arbitrary root systems, Canad. Math. Bull. 53 (2010), 757-762, arXiv:math.CO/0611328.
  25. Woo A., Yong A., Governing singularities of Schubert varieties, J. Algebra 320 (2008), 495-520, arXiv:math.AG/0603273.

Previous article  Next article  Contents of Volume 17 (2021)