### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 495-520 |

Number of pages | 26 |

Journal | Journal of Algebra |

Volume | 320 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2008 |

Externally published | Yes |

### Keywords

- Determinantal ideals
- Interval pattern avoidance
- Kazhdan-Luzstig polynomials
- Schubert varieties
- Singularities

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*320*(2), 495-520. https://doi.org/10.1016/j.jalgebra.2007.12.016