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Interval Pattern Avoidance for Arbitrary Root Systems

Published online by Cambridge University Press:  20 November 2018

Alexander Woo
Alexander Woo*
Affiliation:
Department of Mathematics, Statistics and Computer Science, St. Olaf College, Northfield, MN, U.S.A. e-mail: woo@stolaf.edu
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Abstract

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We extend the idea of interval pattern avoidance defined by Yong and the author for ${{S}_{n}}$ to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for $\text{G}{{\text{L}}_{n}}$ , interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.

Keywords

14M1505E15

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 757 - 762
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bergeron, N. and Sottile, F., Schubert polynomials, the Bruhat order, and the geometry of flag manifolds. Duke Math. J. 95(1998), 373–423. doi:10.1215/S0012-7094-98-09511-4Google Scholar
[2] Billey, S., Pattern avoidance and rational smoothness of Schubert varieties. Adv. Math. 139(1998), 141–156. doi:10.1006/aima.1998.1744Google Scholar
[3] Billey, S. and Braden, T., Lower bounds for Kazhdan–Lusztig polynomials from patterns. Transform. Groups 8(2003), 321–332. doi:10.1007/s00031-003-0629-xGoogle Scholar
[4] Billey, S. and Postnikov, A., Smoothness of Schubert varieties via patterns in root systems. Adv. in Appl. Math. 34(2005), 447–466. doi:10.1016/j.aam.2004.08.003Google Scholar
[5] Billey, S. and Warrington, G., Maximal singular loci of Schubert varieties on SL(n)/B. Trans. Amer. Math. Soc. 355(2003), 3915–3945. doi:10.1090/S0002-9947-03-03019-8Google Scholar
[6] Björner, A. and Brenti, F., Combinatorics of Coxeter groups. Graduate Texts in Math. 231, Springer-Verlag, New York–Heidelberg, 2005.Google Scholar
[7] Bousquet-Mélou, M. and Butler, S., Forest-like permutations. Ann. Comb. 11(2007), no. 3–4, 335–354. doi:10.1007/s00026-007-0322-1Google Scholar
[8] Braden, T. and Macpherson, R., From moment graphs to intersection cohomology. Math. Ann. 321(2001), 533–551. doi:10.1007/s002080100232Google Scholar
[9] Brion, M., Lectures on the geometry of flag varieties. In: Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, 33–85.Google Scholar
[10] Cortez, A., Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire. Adv. Math. 178(2003), 396–445. doi:10.1016/S0001-8708(02)00081-6Google Scholar
[11] Irving, R., The socle filtration of a Verma module. Ann. Sci. École Norm. Sup. Sér. 4 21(1988), 47–65.Google Scholar
[12] Kassel, C., Lascoux, A. and Reutenauer, C., The singular locus of a Schubert variety. J. Algebra 269(2003), 74–108. doi:10.1016/S0021-8693(03)00014-0Google Scholar
[13] Kazhdan, D. and Lusztig, G., Representations of Coxeter Groups and Hecke Algebras. Invent. Math. 53(1979), 165–184. doi:10.1007/BF01390031Google Scholar
[14] Lakshmibai, V. and Sandhya, B., Criterion for smoothness of Schubert varieties in SL(n)/B. Proc. Indian Acad. Sci. Math. Sci. 100(1990), 45–52. doi:10.1007/BF02881113Google Scholar
[15] Lenart, C., Robinson, S. and Sottile, F., Grothendieck Polynomials via permutation patterns and chains in the Bruhat order. Amer. J. Math. 128(2006), 805–848. doi:10.1353/ajm.2006.0034Google Scholar
[16] Manivel, L., Le lieu singulier des variétés de Schubert. Internat. Math. Res. Notices 16(2001), 849–871.Google Scholar
[17] Polo, P., Construction of arbitrary Kazhdan–Luzstig polynomials in symmetric groups. Represent. Theory 3(1999), 90–104 (electronic). doi:10.1090/S1088-4165-99-00074-6Google Scholar
[18] Springer, T., Linear algebraic groups. Second edition, Birkhäuser, Boston, MA, 1998.Google Scholar
[19] Richardson, R. W., Intersections of double cosets in algebraic groups. Indag. Math. (N.S.) 3(1992), 69–77. doi:10.1016/0019-3577(92)90028-JGoogle Scholar
[20] Woo, A. and Yong, A., When is a Schubert variety Gorenstein?, Adv. Math. 207(2006), 205–220. doi:10.1016/j.aim.2005.11.010Google Scholar
[21] Woo, A. and Yong, A., Governing singularities of Schubert varieties. J. Algebra 320(2008), no. 2, 459–520. doi:10.1016/j.jalgebra.2007.12.016Google Scholar