Pattern characterization of rationally smooth affine Schubert varieties of type A

TL;DR

This paper characterizes rationally smooth affine Schubert varieties of type A using pattern avoidance, extending classical smoothness criteria from finite to infinite dimensional affine cases.

Contribution

It introduces a pattern avoidance framework for affine permutations and characterizes rational smoothness in affine Schubert varieties, including twisted spiral permutations.

Findings

Rational smoothness characterized by pattern avoidance in affine permutations

Extension of classical smoothness criteria to affine Kac-Moody groups

Identification of twisted spiral permutations as key elements

Abstract

Schubert varieties in finite dimensional flag manifolds G/P are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai-Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac-Moody group and W being an affine Weyl group or parabolic quotient. In this paper we study the case when W is the affine Weyl group of type A or the affine permutations. We develop the notion of pattern avoidance for affine permutations. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.