TY - JOUR

T1 - Governing singularities of Schubert varieties

AU - Woo, Alexander

AU - Yong, Alexander

N1 - Funding Information:
We thank M. Haiman and F. Sottile for (separately) suggesting that Bruhat-restricted pattern avoidance should have a geometric explanation, inspiring the present study. In addition, we thank S. Billey, J. Carrell, A. Cortez, A. Knutson, V. Lakshmibai, I. Lankham, E. Miller, V. Reiner and G. Smith for helpful discussions. This paper was prepared in part during the authors’ residence at the 2005 AMS Summer Research Institute on Algebraic Geometry in Seattle. In addition, this work was partially completed while A.Y. was an NSERC supported member of the Fields’ Institute during 2005–2006, and while an NSF supported visitor at the Mittag-Leffler Institute during Spring 2005. A.W. was supported in part by NSF VIGRE grant DMS-0135345.

PY - 2008/7/15

Y1 - 2008/7/15

N2 - We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For "reasonable" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P ="singular"; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL (n) / B, Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45-52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205-220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL (n) / B, Trans. Amer. Math. Soc. 335 (2003) 3915-3945, MR 2004f:14071; A. Cortez, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003) 396-445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74-108, MR 2005f:14096; L. Manivel, Le lieu singulier des variétés de Schubert, Int. Math. Res. Not. 16 (2001) 849-871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan-Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.

AB - We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For "reasonable" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P ="singular"; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL (n) / B, Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45-52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205-220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL (n) / B, Trans. Amer. Math. Soc. 335 (2003) 3915-3945, MR 2004f:14071; A. Cortez, Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003) 396-445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74-108, MR 2005f:14096; L. Manivel, Le lieu singulier des variétés de Schubert, Int. Math. Res. Not. 16 (2001) 849-871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan-Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.

KW - Determinantal ideals

KW - Interval pattern avoidance

KW - Kazhdan-Luzstig polynomials

KW - Schubert varieties

KW - Singularities

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U2 - 10.1016/j.jalgebra.2007.12.016

DO - 10.1016/j.jalgebra.2007.12.016

M3 - Article

AN - SCOPUS:44649120024

SN - 0021-8693

VL - 320

SP - 495

EP - 520

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -